Probability distortion, asset prices, and economic growth

Probability distortion, asset prices, and economic growth

Probability Distortion, Asset Prices, and Economic Growth Journal Pre-proof Probability Distortion, Asset Prices, and Economic Growth Maik Dierkes, ...

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Probability Distortion, Asset Prices, and Economic Growth

Journal Pre-proof

Probability Distortion, Asset Prices, and Economic Growth Maik Dierkes, Stephan Germer, Vulnet Sejdiu PII: DOI: Reference:

S2214-8043(18)30477-4 https://doi.org/10.1016/j.socec.2019.101476 JBEE 101476

To appear in:

Journal of Behavioral and Experimental Economics

Received date: Revised date: Accepted date:

13 October 2018 29 July 2019 26 September 2019

Please cite this article as: Maik Dierkes, Stephan Germer, Vulnet Sejdiu, Probability Distortion, Asset Prices, and Economic Growth, Journal of Behavioral and Experimental Economics (2019), doi: https://doi.org/10.1016/j.socec.2019.101476

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Inc.

Highlights • We link potentially irrational preferences to future economic growth.

• Our conjecture is that more irrationality leads to worse allocations of real resources and, ultimately, breeds lower GDP growth in the long run. • We estimate risk preferences via an asset pricing model with Cumulative Prospect Theory agents and distill a recently proposed irrationality index. • Our irrationality index predicts future welfare losses. Predictability is stronger and more reliable over longer horizons. • We find that aggregate preferences show a significant negative impact on future GDP growth if preferences deviate from expected utility as the rational benchmark.

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Probability Distortion, Asset Prices, and Economic Growth Maik Dierkes∗a , Stephan Germera and Vulnet Sejdiua a

Institute of Banking and Finance, Leibniz University Hannover, K¨ onigsworther Platz 1, 30167 Hannover, Germany.

October 4, 2019

Abstract In this paper, we link stock market investors’ probability distortion to future economic growth. The empirical challenge is to quantify the optimality of today’s decision making to test for its impact on future economic growth. Fortunately, risk preferences can be estimated from stock markets. Using monthly aggregate stock prices from 1926 to 2015, we estimate risk preferences via an asset pricing model with Cumulative Prospect Theory (CPT) agents and distill a recently proposed probability distortion index. This index negatively predicts GDP growth in-sample and out-of-sample. Predictability is stronger and more reliable over longer horizons. Our results suggest that distorted asset prices may lead to significant welfare losses.

Keywords: Economic growth, probability distortion, suboptimal decision making JEL Classification: G02, G12

We are particularly grateful to the Jackst¨ adt Foundation for financial support. We thank an anonymous referee, Stefan Trautmann (the Editor), Florian Weigert, Giuliano Curatola, Ivalina Kalcheva, participants at the Swiss Finance Conference 2016, the German Finance Association Annual Meeting 2016, the Research in Behavioral Finance Conference 2016, and the Financial Management Association Annual Meeting 2018 for valuable comments and suggestions. ∗ Corresponding author. E-mail addresses: [email protected] (M. Dierkes), [email protected] (S. Germer), [email protected] (V. Sejdiu).

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Introduction Adam Smith’s concept of the invisible hand promises an efficient allocation of resources in an

economy. The key mechanism behind this efficient allocation is that prices for goods are set to match demand and supply – a cornerstone concept of modern economics. The good’s price ensures an efficient allocation such that it is used for projects with superior profitability. Former Governor of the Federal Reserve System Frederic S. Mishkin vividly stresses the role played by the financial system in this allocation process: “[...] think of the financial system as the brain of the economy: That is, it acts as a coordinating mechanism that allocates capital, the lifeblood of economic activity, to its most productive uses by businesses and households. If capital goes to the wrong uses or does not flow at all, the economy will operate inefficiently, and ultimately economic growth will be low.” [Mishkin (2006)] Mishkin’s implicit assumption that the financial system – or the economy’s brain – is working in a fully rational and efficient way is apparent. Interpreting the financial system as a promoter of economic growth has a long tradition since Bagehot (1873) and Schumpeter (1911) and received some early empirical support by, e.g., Goldsmith (1969). Levine (2005) gives an excellent review of the literature on the finance-growth-nexus. In the present paper, we assume that prices on financial market are set by agents who deviate from the hyper-rational expected utility maximizing agent. Our goal is to analyze the impact of a less than optimally functioning brain – in Mishkin’s sense – on future GDP growth. Specifically, we assume that agents are Cumulative Prospect Theory (CPT) investors and determine prices as in the Barberis and Huang (2008) model. We infer investors’ probability distortion from stock prices and study the impact of probability distortion on future economic growth. Our key finding is that stronger probability distortion today reliably predicts lower future GDP growth in-sample and out-of-sample. This negative link is stronger (i.e. more negative coefficient) and statistically more reliable (smaller p-values and higher adjusted R2 s) over longer prediction horizons. We focus on CPT’s probability distortion when estimating preference parameters from the financial market. A sensitivity analysis reveals that changes in probability weighting has by far the strongest impact on expected returns whereas diminishing value sensitivity and loss aversion only 1

have a second order effect. In similar spirit, Eraker and Ready (2015) suggest that varying probability weighting is more plausible than other preference ingredients when fitting the Barberis and Huang (2008) model to stocks traded over the counter. An implicit assumption of our estimation technique is that risk preferences can vary over time. Guiso et al. (2018) find time varying risk aversion in a large survey among an Italian bank’s clients. They reject changes in wealth or expected income as explanations; more likely, perceived probabilities and emotions play a role. In the lab, Birnbaum (1999), Gl¨ockner and Pachur (2012), and Zeisberger et al. (2012) find that CPT preferences are not stable over time. Campos-Vazquez and Cuilty (2014) find, for example, that anger reduces loss aversion by 50%. Loewenstein et al. (2001), Rottenstreich and Hsee (2001), and Kilka and Weber (2001) identify emotions, affect, and perceived self-competence, respectively, as potential factors driving probability weighting. Wakker (2010) acknowledges that “probability weighting is a less stable component than outcome utility” (p. 228). In the market environment, Polkovnichenko and Zhao (2013) and Dierkes (2013) estimate monthly probability weighting functions from option markets and find substantial variation over time. In particular, probability weighting can survive fierce market competition, large incentives, and environments populated by professional investors with ample experience (see also Post et al., 2008; Sonnemann et al., 2013). To facilitate our predictive regressions, we propose a new index of probability distortion that is able to appropriately quantify deviations from additive probabilities (like in Expected Utility Theory, EUT). It turns out that likelihood insensitivity is a better concept than Tversky and Kahneman’s (1992) probability weighting parameter γ. This concept has been around since Tversky and Wakker (1995) and Tversky and Fox (1995). In the context of inverse S-shaped probability weighting functions, Tversky and Wakker (1995) discuss subadditivity (SA) of decision weights induced by probability weighting. They claim that their proposed “more-SA-than relation can be interpreted as an ordering by departure from rationality” (p. 1266). Hence, we shall use deviations of likelihood sensitivity from additive probabilities in our predictive regressions. Intuitively, likelihood insensitivity captures the idea that, for example, an increase in moderate winning probabilities from 35% to 36% is evaluated as less than a one percentage point increase. At the same time, the decision maker is overly sensitive to changes in small probabilities of extreme events. For example, typical inverse S-shaped probability weighting functions imply likelihood insensitivity whereas oversensitivity is generated by S-shaped weighting functions. Wakker (2010) formalizes 2

an estimation of likelihood insensitivity which we use in our empirical study. Abdellaoui et al. (2011) apply likelihood insensitivity to the domain of decision making under uncertainty. Dimmock et al. (2016) relate it to limited stock market participation. Baillon et al. (2017) use it to explain asked prices on IPO options. Baillon et al. (2018) relate it to health prevention decisions. ˚ Astebro et al. (2015) estimate likelihood insensitivity in the lab and find that it considerably contributes to skewness seeking. Koster and Verhoef (2012) analyze travelers’ misconception of probabilities with likelihood insensitivity. Kilka and Weber (2001) mention it as “source sensitivity” when approximating weighting functions linearly.1 Our paper complements the literature by distilling probability distortion from aggregate stock returns instead of using stock returns directly to predict future GDP growth. Cochrane’s (1991) production-based asset pricing model suggests that variation in stock returns can drive production and real investment activity. Daniel et al. (2002) argue, however, that “large resource misallocations” (p. 174) can result from such a model if equity prices are driven by behavioral factors different from the often hypothesized rational agent’s properties. While our focus is on the deviation from EUT’s independence axiom, Peress (2014) discusses the welfare effects of investor inattention and overconfidence – both clear features of suboptimal investor behavior. Surprisingly, he finds that overconfidence modestly increases income per capita and economic growth in his calibrated model. The logic behind his result is that overconfidence leads to more aggressive trading which improves impounding private information in prices and thus economic welfare. In line with intuition, however, investor inattention leads to lower economic growth due to less capital efficiency. Korniotis and Kumar (2011) show that low investor sophistication can impede risk sharing via financial markets and thereby has detrimental effects on the macro-economy. We focus on CPT’s probability weighting and leave a more detailed analysis of other biases (e.g. overconfidence and other types of probability misestimation such as overoptimism) to future work. Further, we cannot guarantee that our probability distortion estimate does not pick up information about other related biases to some extent. Some unreported results which we discuss in more detail below suggest, for example, that overoptimism (or overpessimism) is closely related to likelihood insensitivity, but a much less strong driver of future GDP growth. It seems that likelihood (in-)sensitivity captures some of the information in a similarly derived optimism/pessimism index. 1

In a different context, Ebert and Prelec (2007) use this insensitivity measure for time preferences.

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Also, we do not provide explicit evidence on the the mechanics which link probability distortion, asset prices, and future GDP growth. Misallocation of scarce resources is a likely channel – as suggested at the beginning of this paper. The internet bubble might serve as a telling example. Internet stocks became overpriced (Green and Hwang, 2011), enabling those overfinanced firms to acquire real resources (e.g. highly educated employees) at inefficiently high prices. Effectively, this led to a redistribution of real resources to the internet industry. Ultimately, however, internet firms failed to show superior profitability on average and the internet bubble burst. Bad decisions about how to allocate resources were made during the 1990s and, years later, hit the economy with a recession in 2001. However, other channels regarding over- or underinvestment, suboptimal savings and consumption decisions which call for adjustments in the real economy, behaviorally driven monetary policy decisions (see e.g. Malmendier et al., 2016) which might interfere with probability distortion, or other explanations are obviously also possible. A detailed analysis of the CPT preference-GDP channel, however, is beyond the scope of this paper. Obviously, there are more determinants of future GDP growth which are related to financial markets. Allen et al. (2012) propose a measure of systemic risk in the banking sector (CATFIN) that, via a lending channel, predicts GDP growth. A series of papers including Fama (1990) and Schwert (1990) find that stock returns positively predict future real activity measured by, for example, production growth rates. Ritter (2005) makes a case for a negative relation between stock returns and GDP growth. Stock and Watson (2003) provide a survey about asset prices and output growth. They find limited predictive ability of stock returns. Liew and Vassalou (2000) and Vassalou (2003) find that the size premium (SMB), value premium (HML), and momentum factor (MOM) have predictive power for GPD growth. Various papers like Officer (1973), Schwert (1989) and Campbell et al. (2001) document countercyclical behavior of stock market volatility and its ability to forecast aggregate GDP growth. Bloom (2009) and Jurado et al. (2015) highlight the importance of macroeconomic uncertainty shocks for future business cycle innovations. When including these variables as controls, our likelihood sensitivity based measure of probability distortion still significantly predicts future GDP growth. Put differently, our probability distortion index is not tantamount to macroeconomic uncertainty measures, systemic risk, or other financial market factors. Finally, our results are robust to numerous variations. These include in-sample and out-of-sample analyses, different calibration procedures of the asset pricing model (simple average returns vs. 4

moving average estimators; GARCH vs. EGARCH), different measures for probability distortion (likelihood insensitivity and Prelec’s (1998) probability weighting function), and sample splits (1953-1984 and 1985-2015). The paper is organized as follows. Section 2 outlines the asset pricing model. Section 3 explains our calibration of the asset pricing model and the estimation of probability distortion in the economy. The data used for this calibration is described in Section 4. Main results are shown in Section 5, corroborated by a battery of robustness checks. Finally, we conclude with Section 6. An appendix provides extensive comparative statics for CPT preference parameters’ impact on the asset prices.

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Sharpe Ratios in an Equilibrium Asset Pricing Model with CPT Investors Here, we recap the main parts of the asset pricing model in Barberis and Huang (2008) and

derive a new theoretical result which relates Tversky and Kahneman’s (1992) CPT preferences in that model to the Sharpe Ratios of the market portfolio. The market portfolio is the value weighted average of all traded securities. In empirical studies, it is typically identified with the value weighted average of all stocks traded in the US (i.e. NYSE, AMEX or NASDAQ). The Sharpe Ratio of the market portfolio is defined as the market portfolio’s excess return, µeM , divided by the standard deviation of the market portfolio’s return, σM . Barberis and Huang (2008) assume that all investors have homogeneous CPT preferences. CPT is widely considered as one of the most powerful descriptive models on individual decision making under risk and uncertainty. It captures experimental evidence such as reference dependence, diminishing value sensitivity, loss aversion and probability weighting (cf. for example Kahneman and Tversky (1979) and Tversky and Kahneman (1992)). e with a continuous Under CPT, an individual evaluates a lottery L which features outcomes x

e in excess of a reference point x∗ using the value function distribution as follows. She evaluates x

v(x) =

   xα

  −λ(−x)β

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x≥0 x<0

(1)

e − x∗ . Diminishing value sensitivity is captured by α ∈ (0, 1) for gains (i.e. positive where x := x

x) and β ∈ (0, 1) for losses (negative x). In particular, the value function is concave for gains and

convex for losses. The parameter λ > 1 models the strength of the decision maker’s aversion to losses. In addition to the value function, CPT features a probability weighting function w+ (P ) =

Pγ (P γ + (1 − P )γ )1/γ

,

w− (P ) =

Pδ . (P δ + (1 − P )δ )1/δ

(2)

which distorts the (de-)cumulative distribution function P of lottery L. In particular, w− (P ) distorts the part of L’s cumulative distribution function associated with losses and w+ (P ) distorts the part of L’s decumulative distribution function associated with gains. γ and δ govern the curvature of the respective probability weighting function. A lower γ (or δ) leads to more overweighting of small probabilities for extreme outcomes. The value γ = 0.28 is interesting because it is the lowest value that produces an increasing probability weighting function (see Camerer and Ho (1994), Rieger and Wang (2006), or Ingersoll (2008)). An increasing probability weighting function is crucial to avoid stochastic dominance violations. γ = 1 is the special case of no probability weighting at all. The overall CPT value assigned to lottery L is calculated as V (L) =

Z 0

−∞

v(x) dw (P (x)) − −

Z ∞ 0

v(x) dw+ (1 − P (x)) .

(3)

Thanks to the extension of the original version of Prospect Theory (see Kahneman and Tversky (1979)) to Tversky and Kahneman’s (1992) cumulative version, CPT can be readily applied to continuous distributions. Based on lab experiments, Tversky and Kahneman (1992) estimate the following values for the parameters of the value function and the weighting function: α = 0.88, β = 0.88, λ = 2.25, γ = 0.61, and δ = 0.69. For parsimony and following existing literature, we henceforth assume α = β and γ = δ. Therefore, preferences are identified with the triple (α, λ, γ). In particular, we write w = w− = w+ . Although many papers assume CPT preferences with these parameter estimates, there is ample evidence that CPT preferences can change over time. In the lab, this has been found by Birnbaum (1999), Gl¨ ockner and Pachur (2012), and Zeisberger et al. (2012). Rottenstreich and Hsee (2001) and Kilka and Weber (2001) find that affect and perceived self-

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competence, respectively, are systematic factors that determine probability weighting. Loewenstein et al. (2001) is yet another source, emphasizing emotions as a key driver of risk preferences. Similar to our approach, Polkovnichenko and Zhao (2013) estimate CPT preferences on a monthly basis from financial markets. Their estimates show considerable fluctuation over time (see e.g. their Figure 6). These are unlikely to be driven by noise or learning over time because they come from one of the most competitive environments with large incentives and they are based on hundreds of option prices and S&P 500 returns set by professional traders. For fixed CPT parameters and given standard deviation σM , the Barberis and Huang (2008) model tells us how to calculate the expected excess return of the market portfolio µeM as follows. Investors derive their one-period CPT value from final wealth relative to today’s wealth compounded with the risk-free interest rate. Effectively, as shown by Barberis and Huang (2008), this is the same e which is assumed as deriving utility from the aggregate stock market’s random excess return rM 2 . Following Barberis and to be conditionally normally distributed with mean µeM and variance σM

Huang (2008, p. 2071ff.), we rewrite Equation (3) as e CP T (rM )=−

Z 0

−∞

 

w Φ

r − µeM σM



dv(r) +

Z ∞ 0



w 1−Φ



r − µeM σM



dv(r),

(4)

where Φ(·) denotes the cumulative distribution function of the standard normal distribution. The CPT value in Equation (4) depends on the expected excess return µeM , the return volatility σM , e ) = CP T (µe , σ , α, γ, λ). and preference parameters α, γ, and λ. We therefore write CP T (rM M M

With conditionally normally distributed returns and preference parameters α ∈ (0, 1), λ > 1,

and γ > 0.28, individuals are µ-σ investors, as Barberis and Huang (2008) show. Thus, all investors choose an efficient portfolio which consists of a combination of the risk-free investment and the

tangency portfolio which, in equilibrium, is the market portfolio. Pricing therefore automatically follows the classical Capital Asset Pricing Model for normally distributed returns. Barberis and Huang (2008) argue that, in equilibrium, the CPT value is zero, i.e. !

CP T (µeM , σM , α, γ, λ) = 0.

(5)

Equation (5) is key to our attempt to reverse engineer CPT preferences from monthly market

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returns in Section 5. We now derive a new result. We show that, in equilibrium, the market portfolio’s Sharpe Ratio depends on preference parameters only, i.e. µeM = f (α, γ, λ) σM

(6)

with f (α, γ, λ) > 0. This result is valid for a piecewise power value function, assuming diminishing value sensitivity is equal for gains and losses (α = β). In particular, our result applies to any monotonically increasing continuous probability weighting function. This insight is new to the literature and interesting on its own. While in Merton (1973) the ratio of equity premium and variance is driven by relative risk aversion, in the Barberis and Huang (2008) framework it is the market portfolio’s Sharpe Ratio that is determined by risk preferences. In particular, higher equity premia result from CPT preferences because of first order risk aversion (Segal and Spivak, 1990). To verify Equation (6), substitute r/σM = x in Equation (4) and use the parametric form of the value function v with α = β as given in Equation (1) to derive e CP T (rM )

Z 0

 

r − µeM =− w Φ σM −∞ =

α σM



· − |

Z 0

−∞

 



w Φ x−

dv(r) +

 µeM

σM

Z ∞ 0



r − µeM w 1−Φ σM

dv(x) + f∗





Z ∞

{z

0

µe α,γ,λ, σM M







dv(r)

µe w 1−Φ x− M σM 





dv(x) . }

!

e ) = 0, we yield an implicit function depending only on the preference In equilibrium, where CP T (rM



µe

M parameters and the equilibrium Sharpe Ratio: f ∗ α, γ, λ, σM



= 0. Hence, the market portfolio’s

Sharpe Ratio is determined by the CPT preference parameters. Although f (·) is not given in analytical form, Equation (6) eases the computational burden considerably. For example, 0.5 is the calculated equilibrium Sharpe Ratio with CPT preference parameters α = 0.88, λ = 2.25, and γ = 0.65. This is consistent with Barberis and Huang’s (2008) finding that a market volatility of 15% leads to an excess market return of 7.5%. According to Equation (6), we can now easily provide more equilibrium examples under these preference parameters: the expected excess market return is half the market volatility. 8

It turns out that changes in the probability weighting parameter have by far the strongest impact on the equity premium. Early research using Prospect Theory focused on loss aversion (and framing) to explain high equity premia (e.g. Benartzi and Thaler (1995) or Barberis et al. (2001)). In the Appendix, however, we provide comparative statics showing that probability weighting is the key driver for expected market returns. We find that the Sharpe Ratio (and hence the expected excess return for a given volatility) of the market portfolio is far more sensitive to changes in the shape of the probability weighting function than to changes in loss aversion or changes in diminishing value sensitivity. Interestingly, probability weighting is at the same time the preference ingredient that relaxes EUT’s independence axiom - often considered a key axiom of rational behavior. For the following estimation of preference parameters from stock returns, it is therefore reasonable to focus on the estimation of γt over time t before contemplating αt or λt (see Eraker and Ready (2015, p. 501) for a similar proposal). When discussing our empirical results in Section 5, we will thus focus on probability distortion. This approach is also consistent with Wakker’s (2010, p. 228) observation that “[i]n general, probability weighting is a less stable component than outcome utility. [. . .] Its shape is, then, influenced by details of perception, and concerns a volatile phenomenon. Therefore, findings on probability weighting cannot be expected to be as stable as findings on utility curvature.” In other words, changes in probability weighting are a likely source for the observed variation in the market portfolio’s Sharpe Ratios.

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Estimation of Probability Distortion from Realized Market Returns In this section, we explain in detail how we estimate CPT’s probability weighting function

and the market portfolio’s Sharpe Ratios. We then motivate an aggregate index of misperceived probabilities based on estimated probability weighting functions which we will use in Section 5 to predict future GDP growth. We use Equation (5) to infer CPT preferences from Sharpe Ratios of the market portfolio. Specifically, we estimate the monthly expected excess return µeMt and the monthly volatility σMt of the market portfolio according to standard procedures in the finance literature as described below. First, we empirically estimate the equity premium µeMt in month t. The (log) equity premium is 9

defined as the difference between the expected (log) return of the market portfolio and the (log) return of a zero-risk investment for a certain investment horizon τ : Equity P remiumt = Et (rMt+τ ) − rft+τ

(7)

with rft+τ being the continuously compounded risk-free rate for the investment period τ , which is already known at time t. Henceforth, we write rft for brevity. In our case, τ equals one month. To estimate the equity premium, we opt for the simplest model possible, i.e. we use the average of historical returns. Welch and Goyal (2008) show that it is hard to beat this simple model, especially when out-of-sample performance is considered.2 To avoid a look-ahead bias in expected returns, we condition on information available up to month t and refer to our estimates µ ˆeMt as conditional equity premium. Specifically, we compute the equity premium as µ ˆeM,t

:= µ ˆMt − rft =

t 1X rM t i=1 i

!

− rft .

(8)

This recursive estimator uses ever more historical returns as t reaches more recent dates – very much like what more sophisticated investors would do. Except for a few months, we yield a positive risk premium and, hence, a positive Sharpe Ratio estimate. A positive equity premium is in line with the well agreed upon notion that aggregate preferences display risk aversion. Second, we estimate the stock market volatility σMt at month t. We use the Generalized Autoregressive Conditional Heteroskedasticity Model (GARCH) of Bollerslev (1986). It captures volatility clustering in return time series and provides a simple way to forecast the next period’s volatility. More precisely, we use a GARCH(1,1) model for parsimony and its wide acceptance in the finance literature on return time series.3 In mathematical terms, we use rMt = µ ˆMt + σMt zt ,

with zt ∼ N (0, 1) i.i.d.

2 2 σM = η + a · (rMt−1 − µ ˆMt−1 )2 + b · σM t t−1

2

(9) (10)

An earlier version of this paper included robustness checks which additionally considered moving average estimators instead of simple average returns for µeM or EGARCH estimates instead of GARCH estimates for σM . Results yield the same conclusions and are not included here for the sake of brevity. They are available, of course, upon request. 3 We also experimented with EGARCH specifications and found our results to be robust.

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2 is where rMt is the realized market return, µ ˆMt is the expected market return estimate, and σM t

the conditional market variance at month t. We assume that the innovation zt is i.i.d.-standard normally distributed. Equation (9) would describe the typical return generating process if the expected return µMt and volatility σMt had been constant. Equation (10) describes variation in 2 the variance depending on the previous realized return rMt−1 and the previous variance level σM . t−1

The following parameter restrictions must hold: η > 0, a ≥ 0, b ≥ 0 and a + b < 1.4 We estimate the parameter vector (η, a, b) for each time point t with the sample of historic market returns rMi , i = 1, . . . , t and using the maximum-likelihood method.5 The log-likelihood function is 

t t t 1X 1X rMi − µ ˆ Mi 2 ln L(η, a, b|rM1 , . . . , rMt ) = − ln 2π − ln σM − i 2 2 i=1 2 i=1 σ Mi

2

.

(11)

2 Values σMi are given by Equation (10) where σM is initialized with the unconditional sample 0

variance of returns rM1 , . . . , rMt . Again, similar to the equity premium estimation, we condition on 2 . With estimates (η, a, b, σ 2 ) we can calculate information available up to month t to estimate σM M0 t

a volatility estimate σMt for every month t via Equation (10). Once we have estimates for µeM and σM we can calculate the estimated Sharpe ratio and infer preferences via Equation (5). As argued in Section 2, changes in the market portfolio’s Sharpe Ratio are best explained with changes of the probability weighting parameter γ. Diminishing value sensitivity α and loss aversion λ only have a second order effect. For example, α appeared rather irrelevant in our comparative statics analysis in the Appendix. In particular, there is less variation in our preference parameter estimates if we focus on changes in γ. Moreover, some analysis of experimental data by Zeisberger et al. (2012) showed virtually no correlation between loss aversion λ and probability weighting γ, which could have confounded our estimation. Additionally, Zeisberger et al. (2012) report in Table 4 that median loss aversion of 73 subjects only changes by 0.05 (from 1.42 to 1.37) over a time span of one month. On the contrary, Polkovnichenko and Zhao (2013) and Dierkes (2013) document substantial changes in aggregate probability weighting inferred from S&P 500 option prices. 4

η 2 The restriction a + b < 1 ensures that the unconditional long-run variance σM = 1−a−b is well defined. Strictly speaking, then, these estimates (η, a, b) have a time index t. Similarly, the unconditional sample variance 2 σM of returns up to time t which is used to initialize the estimation procedure (see below) has a time index t as well. 0 We drop these time indices in our notation for parsimony. 5

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We therefore solve the equilibrium Equation (5) as follows. By varying γ ∈ (0.28, 3) we minimize

the difference between the CPT value on the left hand side and zero on the right hand side of

Equation (5) while keeping α and λ fixed at values estimated by Tversky and Kahneman (1992), i.e. α = 0.88 and λ = 2.25. Afterwards we can adjust λ and α to solve Equation (5). This approach is consistent with Eraker and Ready’s (2015, p. 501) suggestion that varying the probability weighting parameter γ is more plausible when fitting the Barberis and Huang (2008) model to stocks traded over the counter. It is also in line with Wakker’s (2010, p. 228) finding that “probability weighting is a less stable component than outcome utility” (see previous section for more detailed arguments). It turns out that the Tversky and Kahneman (1992) parameter γ itself is not a good proxy for probability distortion. In the context of inverse S-shaped probability weighting functions, Tversky and Wakker (1995) discuss subadditivity (SA) of decision weights as a result of probability weighting. They argue that their proposed “more-SA-than relation can be interpreted as an ordering by departure from rationality” (p. 1266). According to Wakker (2010), this measure of probability distortion is best captured by what he calls likelihood insensitivity. Intuitively, likelihood insensitivity captures the fact that decision makers with the typical inverse S-shaped probability weighting function are not sensitive enough to changes in moderate probabilities while, to the contrary, they are overly sensitive to small probability differences from impossibility or certainty. Technically, the index of likelihood insensitivity equals the slope coefficient Γ of a linear ˆ regression model w(p) = c + Γ · p + p , in which p is a vector (0.01, 0.02, . . . , 0.99) and p is the error term. w(p) ˆ is the empirically estimated probability weighting function of Tversky and Kahneman (1992) with the weighting parameter γˆ estimated as described above. The intuition is that a slope Γ less than one implies that agents do not take an increase of the winning probability from, say, 35% to 36% as a full one percentage point increase, but less than that. They are less sensitive to changes in moderate probabilities than expected utility agents who take probabilities “one for one”. At the same time, they are overly sensitive to small winning probabilities for extreme events. Conversely, a slope larger than one signals oversensitivity to moderate probabilities and insensitivity for small probability changes for extreme events which, again, deviates from the expected utility benchmark agent. Recently, the concept of likelihood insensitivity has become more popular although the idea has been around since Tversky and Wakker (1995) and Tversky and Fox (1995). Abdellaoui et al. 12

(2011) apply likelihood insensitivity to the domain of decision making under uncertainty. Dimmock et al. (2016) relate it to limited stock market participation. Baillon et al. (2017) use it to explain asked prices on IPO options. Baillon et al. (2018) relate it to health prevention decisions. ˚ Astebro et al. (2015) estimate likelihood insensitivity in the lab and find that it considerably contributes to skewness seeking. Koster and Verhoef (2012) analyze travelers’ misconception of probabilities with likelihood insensitivity. Somewhat less related, Ebert and Prelec (2007) use this insensitivity measure in the time domain. In general, the likelihood insensitivity index Γ aggregates and captures the most important features of the Tversky and Kahneman (1992) probability distortion. In particular, Γ = 1 if and only if γ = 1 for our estimates. While Γ < 1 indicates that the probability weighting function is inverse S-shaped, Γ > 1 displays a S-shaped form (underweighting of small probabilities and overweighting of moderate and high probabilities). The more Γ deviates from one, the more the perception of probabilities deviates from the rational expected utility benchmark agent. Therefore, we will henceforth use (Γ − 1)2 as an index of probability distortion when predicting future GDP

growth in Section 5. According to Wakker (2010, p. 229) and Abdellaoui et al. (2011, p. 704), likelihood insensitivity is also closely related to the curvature parameter γP relec of Prelec’s (1998) probability weighting function w(p) = e−(− log(p))

γP relec

. In a robustness check, we therefore employ

(γP relec − 1)2 instead of (Γ − 1)2 .

Finally, likelihood insensitivity also captures some notion of optimism or pessimism. Given the

fit of the linear regression through the Tversky and Kahneman (1992) probability weighting function, w(p) ≈ c + Γp, Wakker (2010) proposes (2c + Γ)/2 as optimism index and, similarly, Abdellaoui et al.

(2011) advocate 1 − Γ − 2c as pessimism index which matches definitions in Kilka and Weber (2001). In particular, the concept of likelihood insensitivity is by construction closely related to optimism

and pessimism. While the slope of the linear regression line through the probability weighting function reflects likelihood insensitivity, the intercept of that regression line – together with Γ – drives optimism (or pessimism). More importantly, our measure of the degree of misperceived probabilities, (Γ − 1)2 , and the optimism index are significantly negatively correlated for our monthly

estimates. The correlation coefficient is −0.5911 with 95% confidence interval [−0.6287, −0.5508].

We are thus reluctant to include the optimism (or pessimism) index in our GDP regressions in addition to (Γ − 1)2 . 13

Apparently, (Γ − 1)2 captures more information about future GDP growth than optimism or

pessimism. In unreported univariate regressions, the optimism index does not significantly predict GDP growth 4 to 8 quarters in the future whereas (Γ − 1)2 does (see Section 5). We will thus focus

on misperceived probabilities measured by (Γ − 1)2 . A possible conclusion is that the curvature of the probability weighting function – in particular overweighting of small probabilities which

generates skewness preferences and, for example, might have contributed to the NASDAQ bubble in the late 1990s – is more important to GDP growth because of a resource misallocation than underor overinvestment induced by mere pessimism or optimism, respectively. A detailed analysis of the CPT preference-GDP channel, however, is beyond the scope of the present paper. Our goal is rather to show that behavioral preferences – once suitably distilled from financial market – do have predictive power for GDP growth.

4

Data We collect data from various sources. Return data are taken from Kenneth French’s website and

compiled as in Fama and French (1993, 1996). RM - RF denotes the market excess return which is defined as the difference of the value-weighted return (RM) of all CRSP firms incorporated in the US and listed on NYSE, AMEX or NASDAQ, and the one-month Treasury bill rate (RF). The factor SMB (Small Minus Big) captures the size premium in returns between stocks with a small and large market capitalization. HML (High Minus Low) is the value premium which indicates the average return difference of stocks with high and low ratios of book value of equity to market value of equity. MOM denotes the momentum factor and is constructed as the difference in returns between portfolios of stocks with high previous and low previous returns, respectively. We use monthly data from July 1926 to December 2015 to infer CPT preference parameters. Macroeconomic data include recession dummies taken from the website of the National Bureau of Economic Research (NBER), the seasonally adjusted GDP time series provided by FRED, the Federal Reserve Bank of St. Louis. We follow Liew and Vassalou (2000) and Vassalou (2003) when selecting control variables for our GDP growth time series regression. These control variables include the default yield spread (DEFY), defined as the yield spread between Moody’s BAA and AAA corporate bonds, the term yield spread (TERMY), defined as the yield difference between the 14

10-year and the 1-year Treasury rate and the detrended wealth variable CAY advocated by Lettau and Ludvigson (2001). Data for DEFY and TERMY are taken from the website of the Federal Reserve Bank of St. Louis. The CAY time series is provided by Martin Lettau. Furthermore, we include the measure of systemic risk in the financial sector (CATFIN) by Allen et al. (2012). Additionally, we use the measure of aggregate macro uncertainty (JLN) and aggregate financial uncertainty (JLN.FIN) of Jurado et al. (2015). Data for CATFIN is taken from Turan Bali’s website, data for JLN and JLN.FIN are obtained from Sydney Ludvigson’s website. Data for GDP growth are available from 1947. DEFY data are obtainable from 1919. TERMY data start in April 1953 and the values for the CAY variable are available from 1953 until 2015:Q3. JLN and JLN.FIN data start in July 1960 and CATFIN is available from January 1973. Therefore, the GDP growth regression model of the in-sample analysis runs from 1953:Q3 and ends 2015:Q4. However, if we include JLN and JLN.FIN the sample starts in 1960:Q3 and if we include CATFIN it starts in 1973:Q1. The out-of-sample analysis uses the full length of the GDP data.

5

Results

5.1

In-Sample Results

We first estimate the probability weighting parameter over time using the Sharpe Ratio of the market portfolio before distilling the likelihood insensitivity index. To compute the expected excess market return, we use a historic mean model. The stock market volatility is computed with a GARCH(1,1) model. All our estimates avoid a look-ahead bias. On average, we obtain an annual market return of 9.82% and an annual risk-free-rate of 3.42% for our sample period. The average equity premium amounts to 6.40%. The Sharpe Ratio estimates are used to infer the CPT preference parameter estimates by using Equation (6). We determine the probability weighting function and the likelihood insensitivity index Γ over time, as described in Section 3. Recall that (Γ − 1)2 quantifies aggregate probability distortion.

[Insert Figure 1 about here.] Figure 1 shows the average probability weighting function estimated for the sample from July 15

1926 to December 2015. The dotted lines correspond to ±3 times the pointwise standard errors at each probability. The average probability weighting function shows an inverse S-shape as found

in experiments by psychologists (e.g. Tversky and Kahneman (1992)). The intersection with the identity line is approximately at probability p = 0.074 which is lower than typically proposed by experimental economists. The likelihood insensitivity coefficient of the average probability weighting function is 0.8744 with a standard error of 0.0123. This shows that the average aggregate probability distortion is significant (t-value of -10.21 for a two-sided t-test that Γ equals one). [Insert Figure 2 about here.] Figure 2 depicts a time series of the estimates of parameter Γt . The curvature index time series is rather persistent and provides evidence for long periods of strong probability distortion. In particular, from November 1949 to May 1962, from December 1991 to April 1997, from November 2001 to October 2008, and from July 2009 to December 2015 (sample period end) our model estimates large chunks of months with curvature parameters Γt < 1 (except for a few estimates). In these months, the probability weighting function exhibits the inverse S-shaped form typically found in lab experiments. If the estimated Sharpe Ratio resides at economically implausible (negative) values, our curvature index captures this characteristic and approaches the value Γ = 1.0281 which corresponds to the solver’s limit of 3 set for the Tversky and Kahneman (1992) parameter γ. Negative Sharpe Ratios result during the Great Depression, around the mid 1970s, and early 1980s due to high risk-free rates and low market returns which implies flat Γ estimates. Note that such constant values for Γ do not enhance our explanatory power and suggest that our predictive regression results are conservative. Moreover, Figure 2 confirms the observation of Birnbaum (1999), Gl¨ ockner and Pachur (2012), and Zeisberger et al. (2012) from lab experiments that individuals’ CPT preferences vary over time. The substantial variation over time of aggregate probability distortion is also shown by Polkovnichenko and Zhao (2013) and Dierkes (2013) who used option prices rather than stock prices. [Insert Table 1 about here.] Table 1 provides descriptive statistics for all variables entering the GDP regressions.

16

Next, we turn to our main results. We regress GDP growth on our probability distortion index (Γ − 1)2 to investigate whether a larger deviation from expected utility maximizers due to probability weighting (i.e. Γ 6= 1) yields more negative changes in economic output. Since GDP is reported

quarterly, we use quarterly averages of the probability distortion index, the default yield spread

(DEFY), the term yield spread (TERMY), the aggregate macro uncertainty (JLN) and aggregate financial uncertainty (JLN.FIN) as well as the measure of systemic risk in the financial sector (CATFIN) in the regressions. The detrended wealth variable CAY is already specified on a quarterly basis. The stock market variables are the quarterly excess market return RM - RF, size premium SMB, value premium HML, and momentum MOM factors. Additionally, we use the quarterly stock market volatility M.Vola which is the quarterized estimate of the conditional market volatility σ ˆMt . The motivation for using these controls is as follows. Our methodology distills the investors’ probability distortion from the market portfolio’s excess returns. However, the excess market return itself might contain additional or even more valuable information about future GDP growth. Fama (1990) and Schwert (1990), for example, make a case for this latter reasoning. Hence, we include the quarterly excess market return RM - RF as control. Officer (1973), Schwert (1989) and Campbell et al. (2001) also document an ability of the stock market volatility to forecast business fluctuations and ultimately economic output. Liew and Vassalou (2000) and Vassalou (2003) find that SMB, HML and MOM have ability to predict future GDP growth. They argue that the risk of these return factors is systematically related to the business cycle and, therefore, they carry information about future GDP growth. For example, small market capitalization stocks are more likely to default in the next recession than large market capitalization stock. Hence the size premium SMB (“small minus big”) is positive and can predict future GDP growth. We further follow their suggestion to use the default premium DEFY, term premium TERMY, and detrended wealth CAY as control variables. Allen et al. (2012) and Jurado et al. (2015) document the predictive ability of CATFIN and JLN, respectively, on future economic growth. Note that our probability distortion index is negatively correlated with the macroeconomic uncertainty proxy JLN and also with CATFIN which proxies for systemic risk. The correlation with JLN is −0.259 and with CATFIN it is −0.307. In

particular, our probability distortion index is not merely an alternative measure of uncertainty or risk.

17

[Insert Table 2 about here.] Table 2 summarizes the results of the time series regressions for GDP growth from 1953:Q3 for models (1)–(3) to 2015:Q4. The sample of model (4) starts in 1973:Q1 due to the availability of CATFIN. We group all models into subsections Q0 to Q8 depending on the lag structure of our explanatory variables. In each subsection, model (1) includes only our probability distortion index (Γ − 1)2 . We gradually add the default yield premium DEFY, the term yield premium TERMY, detrended wealth CAY (see model (2)), quarterly returns RM-RF, SMB, HML, MOM and stock

market volatility M.Vola (see model (3)) and, finally, JLN and CATFIN (see model (4)) as controls. In subsection Q0, none of the explanatory variables in models (1) to (4) is lagged. Without lags, adjusted R2 s are increasing with the number of controls where JLN and CATFIN have – not surprisingly – the highest marginal explanatory power, boosting adjusted R2 from 9% to 21%. The loadings on our distortion index are negative as expected and significant in all models, even when JLN and CATFIN are included in model (4). We lag our probability distortion index by two, four, six, and eight quarters in subsections Q2 to Q8. In each lag subsection Q2 to Q8, we consider four models (1) to (4) with gradually adding controls as we did for Q0. However, to give our control variables the best opportunity to explain GDP growth, we lag control variables by only one quarter. This gives our probability distortion index, which is lagged by more than one quarter, the toughest ground to survive in predictive regressions. For all lags, we observe a negative influence of probability distortion on future GDP growth. Interestingly, loadings become more negative for longer prediction horizons. This is consistent with our conjecture that bad decision making today about which projects to finance will materialize in the longer run in lower economic output. Moreover, predictability increases with longer horizons as indicated by lower p-values and higher adjusted R2 s. (Γ − 1)2 is significant for all lags, even

in the presence of business cycle variables, aggregate uncertainty and systemic risk measures, as well as asset pricing factors which are known to be related to future GDP growth (see Liew and Vassalou (2000), Vassalou (2003), Allen et al. (2012), Jurado et al. (2015), Campbell et al. (2001)). The adjusted R2 is highest at lag 8 (0.2808 including and 0.0529 excluding control variables). Panel A in Figure 3 visualizes the decreasing regression coefficients for our probability distortion 18

index when we increase lags from zero to eight quarters. Each block of bars is related to a lag subsection (Q0 to Q8). And each block’s bars depict the regression coefficient’s value from the four models (1) to (4). To gauge statistical significance we add 95%-confidence intervals based on Newey and West (1987, 1994) standard errors. A natural concern is that the financial system’s disruption during the recent financial crisis in 2008–2009 and its aftermath hurts economic output and also biases our probability distortion estimates from stock prices. In order to guard against this concern, we conduct a sample split which employs two subsamples of equal size: (i) the beginning of the sample in each model until 1984:Q3 and (ii) 1984:Q4 – 2015:Q4. For models (1)–(3) our sample starts in 1953:Q3 for the first subsample. Since JLN.FIN and CATFIN are highly correlated (54%) we opt to use JLN.FIN instead of CATFIN for the first subsample in model (4) to increase the number of observations. Note, that JLN.FIN starts in 1960:Q3 while CATFIN data are available from 1973:Q1. The second subsample uses CATFIN in model (4).6 Results are reported in Table 3. [Insert Table 3 about here.] Overall, the patterns for both subsamples confirm our baseline results in Table 2. That is, regression coefficients for (Γ − 1)2 become more negative for higher lags and adjusted R2 s increase. However, statistical significance for the coefficients of the probability distortion index are weaker up to lag Q6. At lag Q8, we yield significant loadings in both subsamples (p < .05 except for model (3) in the early subsample which yields p < .1). Overall, this pattern is in line with our hypothesis. [Insert Figure 3 about here.] Since likelihood insensitivity is closely related to the parameter γP relec in Prelec’s (1998) probability weighting function w(p) = e−δ(− log p)

γP relec

, we check the robustness of our results by

re-running the predictive regressions with (γP relec − 1)2 as an alternative measure of probability distortion in asset prices. We obtain the curvature parameter γP relec when fitting log(− log w(p)) =

log δ + γP relec log(− log p) on the open interval (0,1). Abdellaoui et al. (2011) argue on page 704 that γP relec plays a similar role as the likelihood insensitivity index Γ. 6

We modified model (4) in the second subsample to include JLN.FIN instead of CATFIN and our probability distortion index is still significant at lags Q6 and Q8. Additionally, we also rerun model (4) with JLN.FIN for the whole sample as in Table 2 and again find a negative impact of our probability distortion index on (future) GDP growth.

19

Panel B in Figure 3 presents the corresponding time series regression coefficients for (γP relec − 1)2 .

As noted above, the bars correspond to the regression coefficients of the panel-specific probability distortion index alongside 95% confidence intervals based on Newey and West (1987, 1994) standard

errors. We find a significant negative effect of (γP relec − 1)2 on future GDP growth for all lags.

The coefficients of determination (not reported) indicate slightly stronger results than our main

findings in Table 2 and are again increasing with higher lags. The correlation between our baseline insensitivity index Γ and the curvature parameter γP relec is 0.9879. Hence, our results are robust to the choice of the probability distortion measure. Moreover, our results are also robust to different calibration procedures of the asset pricing model (moving average estimators instead of simple average returns for µeM ; EGARCH estimates instead of GARCH estimates for σM ). We do not report these results for sake of brevity. They are, of course, available upon request.

5.2

Out-of-Sample Results

After having presented in-sample predictability, we now turn to the out-of-sample predictability of (Γ − 1)2 . In the following, we test the out-of-sample performance of (Γ − 1)2 by applying the Clark and West (2007) test for nested models. The predictive regression model is GP DGRt+h = β0 + β1 GDP GRt + β2 (Γt − 1)2 + t+h ,

(12)

where GP DGRt denotes the GDP growth, (Γt − 1)2 the probability distortion index, t the respective

quarter, and h the forecast horizon. In particular, we test the out-of-sample predictive power of model (12) against three nested benchmark models: (1) a classical AR(1)-model (β2 = 0), (2) the random-walk hypothesis (β0 = β2 = 0 and β1 = 1), and (3) the historical average (β1 = β2 = 0).

While the random-walk hypothesis and the historical average are typical benchmarks in the finance literature (e.g. Rapach and Zhou (2013)), Stock and Watson (2003) and Clark and West (2007) argue that the adequate benchmark for GDP growth forecasts is an AR(1). To assess the relative performance of the predictive regression model, we follow Campbell and Thompson (2008) and use the out-of-sample R2 which measures the proportional reduction in mean squared error (MSE) for the regression forecast based on (Γ − 1)2 (M SE1 ) relative to the benchmark forecast (M SE0 ) 20

(Rapach and Zhou, 2013): 2 ROS =1−

M SE1 . M SE0

(13)

The MSE for each model over the out-of-sample period is given by M SEi =

out  2 1 NX \ GDP GRNin +k − GDP GRi,Nin +k , Nout k=1

(14)

\ where Nin and Nout indicate the in-sample and out-of-sample sizes, and GDP GRi,Nin +k the forecast value for the quarter Nin + k. 2 Then, if ROS is significantly greater than zero, we can conclude that our probability distortion

index has also out-of-sample predictability. To test the statistical significance, we compute the 2 Clark and West (2007) MSE-adjusted statistic for testing the null hypothesis that ROS is smaller 2 than or equal to zero against the alternative that ROS is greater than zero. This corresponds to H0 :

M SE0 ≤ M SE1 against H1 : M SE0 > M SE1 . Since we assess multi-step forecasts, we additionally adjust the Clark and West (2007) test statistic for heteroskedasticity and autocorrelation according to Newey and West (1987) where the lag choice equals the respective forecast horizon. We evaluate the out-of-sample predictability of our probability distortion index again for a 2-quarter, 4-quarter, 6-quarter, and 8-quarter forecast horizon. Table 4 presents the results of the out-of-sample analysis. Panel A of Table 4 reports the results of a rolling regression with in-sample size of Nin = 120 (30 years). Panel B shows the results of a recursive regression. The in-sample period starts in 1944:Q2 (begin of GDP data) and the prediction period begins from 1977:Q4 and ends 2015:Q4. Table 4 shows that the predictive regression model based on our probability distortion index 2 > 0) all benchmark models for almost all forecast horizons. (Γ − 1)2 significantly outperforms (ROS

Only the 2-quarter forecast of the AR(1) model produces a lower MSE than the probability distortion

model. But this is in line with our intuition that probability distortion has rather a long-term effect on GDP growth. This conjecture is also confirmed by the fact that the out-of-sample R2 from the comparison with the AR(1) increases with higher forecast horizon (except for the recursive 2 s are at the 10% level significant only, it is still remarkable that regression Q8). Although some ROS

21

a probability distortion index inferred from stock prices beats an AR(1), especially, given the finding that stock prices itself have empirically only limited predictive ability (Stock and Watson, 2003). Overall, we conclude that aggregate preferences show a significant negative impact on GDP growth if preferences deviate from expected utility as the rational benchmark. To the extent that probability distortion survives in financial markets, it poses a threat to economic growth.

6

Conclusion Aggregate risk preferences in an economy determine demand and supply. Normative theories

about economic agents and, in particular, risk preferences imply efficient prices and thus an optimal allocation of scarce real resources. However, the number of papers challenging this traditional view of rationally deciding economic agents is increasing. In this paper, we distill an important factor of suboptimal economic decision making from stock prices. In an equilibrium asset pricing model with CPT investors, probability distortion manifests in stock prices. This non-additive processing of probabilities manifests in agents being overly sensitive or insensitive to likelihoods. After distilling this likelihood (in)sensitivity from the market portfolio’s returns, we show that the deviation of this probability misperception from linear probabilities predicts lower future GDP growth. Predictability is stronger and more reliable over longer horizons. Our conjecture is that suboptimal decision making is one channel by which today’s market prices and future GDP growth are linked. Implicitly, we provide evidence that stock prices can deviate from their rationally warranted fundamental value.

Appendix: Comparative Statics In the following, we show that the expected market return is most sensitive towards changes in probability weighting. Figure A.1 illustrates the influence of changes in the preference parameters on the slope of the capital market line, i.e. the Sharpe Ratio. The left panels provide a threedimensional plot and the right panels depict the corresponding contour lines. We rather use the Sharpe Ratio than the equity premium since then no specification of the market volatility is needed. Using Equation (6), we are able to distinguish between a volatility effect and a preference effect

22

on the equity premium. Thus, a representation of the Sharpe Ratio is more instructive than a representation of the equity premium. [Insert Figure A.1 about here.] First, we discuss the sensitivity of the market portfolio’s Sharpe Ratio to changes to the CPT investor’s diminishing value sensitivity captured by the value function’s parameter α. The influence of α on the Sharpe Ratio of the market portfolio for various λ and γ values is depicted in the middle and lower rows of Figure A.1 (the upper row keeps α fixed at 0.88). Figure A.2 shows the Sharpe Ratio as a function of α for our standard preference parameters γ = 0.65 and λ = 2.25. [Insert Figure A.2 about here.] The relationship between the Sharpe Ratio and α is almost linear. The slope of the function at α = 0.88 equals approximately −0.18. This sensitivity coefficient reveals how much the Sharpe Ratio increases if α increases by, say, 0.01. Assuming σM = 0.15, as in the numerical example

above, this corresponds to a 0.027 percentage point decrease of the equity premium (from 7.5% down to 7.473%) if α ∈ (0, 1) increases by 0.01. This result is robust with respect to the choice of

the parameters γ and λ. Figure A.1 likewise illustrates the impact of α when varying the CPT parameters γ and λ. Interestingly, for α = 1 the value function is piecewise linear and implies a Sharpe Ratio of 0.48 and a market excess return of 7.2%. This is slightly lower than the Sharpe Ratio of 0.5 and the equity premium of 7.5% with the initial preference parameter values of Barberis and Huang (2008). The example demonstrates that the assumption of a concave value function for gains and a convex one for losses is not necessary to observe a substantial equity premium (see also Barberis and Huang (2008)). We turn to the impact of changes in the loss aversion parameter λ on the Sharpe Ratio. An investor evaluates a loss as λ times worse than a profit of the same quantity. Figure A.3 illustrates the functional relationship between the Sharpe Ratio and the loss aversion coefficient λ for α = 0.88 and γ = 0.65 and shows that the relationship is close to linear. [Insert Figure A.3 about here.] 23

The slope of the function at λ = 2.25 equals approximately 0.27. Again, this gives a sensitivity coefficient, which tells us how much the Sharpe Ratio increases if λ increases by one unit. In terms of equity premium, if σM = 0.15, an increase of λ by 0.01 leads to an increase of the equity premium by 0.041 percentage points. Consistent with intuition, higher loss aversion λ results in higher risk compensation. For a representation of the Sharpe Ratio as a function of λ with varying α and γ, see Figure A.1. As noted by Barberis and Huang (2008), the assumption λ > 1 is crucial for a positive equity premium, independent of α and γ. To conclude the comparative statics, we finally examine the effect of the curvature parameter γ of the probability weighting function in Equation (2). This parameter captures the effect that people distort probabilities. On average, they tend to overweight unlikely and extreme events and to underweight events with average or high probability. Again, we set α = 0.88 and λ = 2.25. [Insert Figure A.4 about here.] Figure A.4 reveals a convex functional relationship between the Sharpe Ratio and γ. The Sharpe Ratio is more sensitive to γ than to α or λ. The slope of the function at γ = 0.65 equals approximately −0.76, i.e. in absolute terms, a sensitivity almost three times as high as the sensitivity to changes in loss aversion and more than four times higher than changes in diminishing value

sensitivity. Assuming σM = 0.15, if γ changes from 0.65 to 0.64 the equity premium will increase by 0.114 percentage points. The negative functional relationship between γ and the Sharpe Ratio is due to more overweighting of both tails of the normal distribution for lower γ. Due to loss aversion (λ > 1), overweighting of the lower tail hurts investors more than overweighting of the upper tail enjoys them, which makes them command a higher risk premium. For γ = 1, objective probabilities equal transformed probabilities and there is no longer a weighting effect. In this case the equilibrium Sharpe Ratio is 0.335 and the equity premium is about 5%. This is much lower than 7.5% in the initial case with (α, γ, λ) = (0.88, 0.65, 2.25). [Insert Table A.1 about here.] Table A.1 summarizes the parameter constellations and their impact on the Sharpe Ratio discussed above. We find the following sensitivities of which the probability weighting parameter γ

24

has the highest impact in absolute terms on the slope of the capital market line: µe



M ∂ σM ∂γ

µe

γ=0.65

≈ −0.76,



M ∂ σM ∂λ

µe

λ=2.25

≈ 0.27,



M ∂ σM ∂α

α=0.88

≈ −0.18.

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28

1.0

0.8

0.6 w(p) 0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

p

Figure 1: Average probability weighting function. We estimate 1,074 probability weighting functions from July 1926 to December 2015 as parameterized in Tversky and Kahneman (1992). This figure illustrates ¯ the average probability weighting function w(p) (solid blue line) and ±3 times the empirical pointwise standard error (dotted lines). The gray line corresponds to the identity line.

1.25

1.00

0.75 Γt 0.50

0.25

0.00 1930

1940

1950

1960

1970

1980

1990

2000

2010

Date

Figure 2: Likelihood insensitivity estimates over time. We estimate 1,074 probability weighting functions w(p) ˆ from July 1926 to December 2015 as parameterized in Tversky and Kahneman (1992). This figure depicts the index of likelihood insensitivity (curvature index) which equals the slope coefficient of a linear regression model w(p) ˆ = c + Γ · p + p , in which p equals a sequence {0.01, 0.02, . . . , 0.99} and p is the error term. The gray bars indicate the recessions in the US after 1926.

29

0 −1 −2 −3 −4

Regression coefficient of (γPrelec − 1)2

−6

−5

0 −1 −2 −3 −4

Regression coefficient of (Γ − 1)2

−5 −6

Q0

Q2

Q4

Q6

Q8

Q0

Q2

Lags

Panel A: Baseline Results Model (1) without control variables

Q4

Q6

Q8

Lags

Panel B: Prelec’s Probability Weighting Function

Model (2) with control variables DEFY, TERMY, CAY

Model (3) with control variables DEFY, TERMY, CAY, RM−RF, SMB, HML, MOM, M.Vola

Figure 3:

Model (4) with control variables DEFY, TERMY, CAY, RM−RF,60%, HML, MOM, M.Vola, -/1&$7),1

Factor loadings on probability distortion when predicting future GDP growth. For each Panel and each lag specification (Q0 to Q8), we perform four time series regressions. Firstly, we regress GDP growth on the panel-specific probability distortion index. Secondly, we include DEFY, TERMY and CAY as control variables. Thirdly, we add RM-RF, SMB, HML, MOM and M.Vola as control variables. Finally, we additionally include JLN and CATFIN. Regression coefficient estimates of the probability distortion index are represented as bars. Black lines correspond to 95% confidence intervals based on Newey and West (1987, 1994) standard errors. Panel A recaps regression coefficients of probability distortion from our baseline results as presented in Table 2. In Panel B, the curvature parameter of Prelec’s (1998) probability weighting function is used to calculate the probability distortion index in order to account for an alternative specification of the weighting function. Q0 denotes that no explanatory variable is time-lagged. Q2, Q4, Q6 and Q8 indicate that only our curvature parameter is 2, 4, 6 or 8 quarters lagged whereas the control variables are always lagged by one quarter. For models (1) to (3) the sample starts in 1953:Q3 and ends in 2015:Q4. Note, that for model (4) the sample starts in 1973:Q1 due to the availability of CATFIN. DEFY is the yield spread between Moody’s BAA and AAA corporate bonds. TERMY denotes the yield spread between long-term government bonds and the T-bill. CAY denotes the detrended wealth. JLN is the measure of economic uncertainty of Jurado et al. (2015). CATFIN is the measure of systemic risk in the financial sector proposed by Allen et al. (2012). Data are obtained from Turan Bali’s, FRED’s, Martin Lettau’s, and Syndey Ludvisgson’s websites. RM - RF, SMB, HML and MOM describe quarterly excess market, size, value and momentum returns, taken from Kenneth French’s data library. M.Vola denotes quarterly stock market volatility.

30

1.0

Sharpe Ratio, α=0.88

0.3

5

0.3

0.9

0.4

0.8

0.7 0.6 0.5

0.5

0.7

mu/sigma

0.45

γ

0.4 0.55

2.5

0.9

0.8 gam 0.7 ma

0.6

2.1

0.6

2.4 2.3 a 2.2 bd m la

0.6

0.65

0.7

0.5

0.3 1.0

0.5 2.0 2.0

2.1

2.2

2.3

2.4

2.5

λ

1.0

Sharpe Ratio, λ=2.25

0.9

0.35

0.6

0.4

γ

0.5

0.7

mu/sigma

0.8

0.7

0.4

1.0

0.5

0.8 gam 0.7 ma

0.6

0.6

0.8 a 0.7 lph a

0.55

0.6

0.5

0.9

0.6

0.9 1.0

0.45

0.75

0.65

0.7

0.5 0.5 0.5

0.6

0.7

0.8

0.9

1.0

α

2.5

Sharpe Ratio, γ=0.65

64

2.4

0.

0.65

62

0.

0.55

λ

0.50 0.45

8

0.5

6 0.5

1.0

4

0.5

2.1

0.9 0.8 2.4

2.3 lamb 2.2 da

2.1

0.6

0.7 lph a

2

0.5

a 0.5

2.0

2.5

0.6

2.2

mu/sigma

2.3

0.60

8

0.4

0.4

6

4 0.4

2 0.4

2.0 0.5 0.5

0.6

0.7

0.8

0.9

1.0

α

Figure A.1: Influence of the CPT preference parameters on the Sharpe ratio in equilibrium. Function f (α, γ, λ) from Equation (6) is used to derive the corresponding slopes of the capital market line. The left panels provide a three-dimensional plot and the right panels depict contour lines. The upper panel assumes α = 0.88, the center panel λ = 2.25 and the lower panel γ = 0.65.

31

Sharpe Ratio, γ=0.65, λ=2.25

0.58

0.56

0.54 µeM σM 0.52

0.50

0.48 0.5

0.6

0.7

0.8

0.88 0.9

1.0

α

Figure A.2: Influence of the CPT preference parameter α on the Sharpe Ratio. This figure shows the Sharpe Ratio as a function of α for the standard preference parameters γ = 0.65 and λ = 2.25. Given this parameter setting, α = 0.88 yields a Sharpe Ratio of 0.5 (green lines) in equilibrium.

Sharpe Ratio, α=0.88, γ=0.65

0.56

0.54

0.52

µeM 0.50 σM 0.48

0.46

0.44

2.0

2.1

2.2

2.25

2.3

2.4

2.5

λ

Figure A.3: Influence of the CPT preference parameter λ on the Sharpe Ratio. This figure shows the Sharpe Ratio as a function of λ for the standard preference parameters α = 0.88 and γ = 0.65. Given this parameter setting, λ = 2.25 yields a Sharpe Ratio of 0.5 (green lines) in equilibrium.

32

Sharpe Ratio, α=0.88, λ=2.25 0.65

0.60

0.55

µeM 0.50 σM 0.45

0.40

0.35

0.5

0.6

0.65

0.7

0.8

0.9

1.0

γ

Figure A.4: Influence of the CPT preference parameter γ on the Sharpe Ratio. This figure shows the Sharpe Ratio as a function of γ for the standard preference parameters α = 0.88 and λ = 2.25. Given this parameter setting, γ = 0.65 yields a Sharpe Ratio of 0.5 (green lines) in equilibrium.

33

Table 1: Descriptive statistics for data entering the GDP predictive regressions. GDP denotes the seasonally adjusted GDP. (Γ − 12 ) is the probability distortion index on a quarterly basis. DEFY is the yield spread between Moody’s BAA and AAA corporate bonds. TERMY denotes the yield spread between long-term government bonds and the T-bill. CAY denotes the detrended wealth. JLN is the aggregate macro uncertainty measure and JLN.FIN the aggregate financial uncertainty measure of Jurado et al. (2015). CATFIN is the measure of systemic risk in the financial sector proposed by Allen et al. (2012). Data are obtained from Turan Bali’s, FRED’s, Martin Lettau’s, and Syndey Ludvisgson’s websites. RM - RF, SMB, HML and MOM describe quarterly excess market, size, value and momentum returns, taken from Kenneth French’s data library. M.Vola denotes quarterly stock market volatility. All statistics are calculated for quarterly data. We use quarterly averages of DEFY, TERMY, JLN, JLN.FIN and CATFIN. GDP growth, DEFY, TERMY, RM - RF, SMB, HML, MOM and M.Vola are given in percent. Std denotes the standard deviation and N stands for the number of observations. Min

1st Quartile

Median

Mean

3rd Quartile

Max

Std

N

Data range

1.04

1.468

1.551

2.116

5.791

1.017

250

1953:Q3 - 2015:Q4

0

0.0012

0.0157

0.0658

0.0875

0.4376

0.0996

257

1951:Q3 - 2015:Q3

DEFY

0.36

0.7

0.865

0.9777

1.157

3.02

0.439

250

1953:Q2 - 2015:Q3

TERMY

-1.94

0.1925

0.87

0.977

1.74

3.35

1.093

250

1953:Q2 - 2015:Q3

CAY

-4.728

-1.26

0.0451

-0.0578

1.422

4.38

1.978

250

1953:Q2 - 2015:Q3

RM - RF

-26.85

-3.12

2.858

1.812

6.895

23.37

8.393

250

1953:Q2 - 2015:Q3

SMB

-12.77

-2.521

0.0206

0.5131

3.806

15.39

5.179

250

1953:Q2 - 2015:Q3

HML

-17.17

-2.375

1.057

1.075

3.576

26.04

5.467

250

1953:Q2 - 2015:Q3

MOM

-39.6

-0.7053

2.107

2.258

5.749

27.99

7.118

250

1953:Q2 - 2015:Q3

M.Vola

3.813

6.028

7.354

7.690

8.871

15.78

2.182

250

1953:Q2 - 2015:Q3

JLN

0.5389

0.5998

0.6364

0.6581

0.6775

1.053

0.0909

221

1960:Q3 - 2015:Q3

JLN.FIN

0.6041

0.7768

0.8887

0.909

1.019

1.496

0.1655

221

1960:Q3 - 2015:Q3

CATFIN

0.1194

0.1961

0.25

0.2682

0.317

0.6614

0.0996

171

1973:Q1 - 2015:Q3

GDP growth -1.975 (Γ − 1)

2

34

35

1.6588

2.3646

2.1308

∗∗∗

250

0.2439

250

0.0095

MOM

0.0141

0.0210

0.0783

0.0902

250

0.2146

171 0.0167

250

(0.9650)

−3.1662∗∗∗

CATFIN

(1.7503)

−3.4909∗∗

JLN

(0.0584) (0.0461)

−0.0527

(0.0097) (0.0107)

0.0104

0.0186 −0.0057

(0.0148) (0.0146)

HML

(0.0157) (0.0156)

0.0056

(0.0109) (0.0118)

(0.0463) (0.0463) (0.0623)

(0.0879) (0.0882) (0.0838)

−0.0652 −0.0648 −0.0640

−0.1353 −0.1240 −0.2073∗∗

(0.4164) (0.3257) (0.4015)

−0.3306 −0.2015

0.0246

adj. R2

1.9949

∗∗∗

(3)

(1) 1.6760

∗∗∗

4.3078

∗∗∗

(4)

(3) 1.9697

∗∗∗

Q4

1.9774

∗∗∗

(2)

(1) 1.7000

∗∗∗

4.1939

∗∗∗

(4)

(3) 2.0580

∗∗∗

Q6

2.0374

∗∗∗

(2)

(1) 1.7161 ∗∗∗

4.1215 ∗∗∗

(4)

(3) 2.1085 ∗∗∗

Q8

2.0900 ∗∗∗

(2)

(4) 3.9487∗∗∗ ∗∗∗

0.1754

0.0535

250

0.0403∗∗∗

0.0050 0.0260

0.1030

250

0.2509

0.0300

250

(1.2769)

171

−3.7420∗∗∗

(2.2139)

−2.6736

(0.0531) (0.0552)

−0.0305

(0.0111) (0.0117)

0.0110

0.0279∗ −0.0092

(0.0148) (0.0121)

(0.0173) (0.0153)

0.0275

(0.0105) (0.0103)

0.0212∗∗−0.0187∗

(0.0462) (0.0447) (0.0665)

(0.0699) (0.0685) (0.0711)

−0.1008∗∗−0.0910∗∗−0.0826

−0.0302 −0.0640 −0.1149

(0.3867) (0.2676) (0.3222)

−0.2415 −0.1276

0.2019

0.0681

250

0.0373∗∗

0.0034 0.0352

0.1152

250

0.2620

0.0428

250

(1.1797)

171

−3.8392∗∗∗

(2.1572)

−2.5662

(0.0470) (0.0508)

−0.0215

(0.0109) (0.0110)

0.0115

0.0261∗ −0.0121

(0.0152) (0.0133)

(0.0175) (0.0156)

0.0287

(0.0101) (0.0098)

0.0206∗∗−0.0186∗

(0.0445) (0.0429) (0.0596)

(0.0646) (0.0625) (0.0705)

−0.0970∗∗−0.0863∗∗−0.0693

−0.0489 −0.0842 −0.1490∗∗

(0.3473) (0.2548) (0.3213)

−0.2311 −0.1385

0.1787

0.0878

250

0.0356∗∗

0.0316

0.1368

250

0.2730

0.0529

250

(1.1139)

171

−3.7020∗∗∗

(2.1271)

−2.3694

(0.0434) (0.0525)

−0.0230

0.0086 −0.0002

(0.0104) (0.0099)

0.0260∗ −0.0100

(0.0150) (0.0126)

(0.0171) (0.0158)

0.0271

(0.0100) (0.0096)

0.0218∗∗−0.0153

(0.0422) (0.0398) (0.0547)

(0.0611) (0.0578) (0.0694)

−0.0993∗∗−0.0894∗∗−0.0663

−0.0682 −0.1044∗ −0.1815∗∗∗

(0.3190) (0.2494) (0.3307)

−0.2432 −0.1564

0.1370

0.1073

250

0.0355∗∗

0.0025 0.0304

0.1553

250

0.2808

171

(1.1141)

(2.3388) −3.5997∗∗∗

−2.0634

(0.0407) (0.0492)

−0.0219

(0.0102) (0.0099)

0.0082

0.0245 −0.0083

(0.0149) (0.0130)

(0.0162) (0.0155)

0.0268∗

0.0218∗∗−0.0115

(0.0095) (0.0094)

(0.0423) (0.0386) (0.0523)

(0.0589) (0.0546) (0.0757)

−0.1049∗∗−0.0948∗∗−0.0527

−0.0890 −0.1247∗∗−0.2024∗∗∗

(0.3144) (0.2537) (0.3563)

−0.2500 −0.1678

(0.7263) (0.7725) (0.9531) (1.1568) (0.7320) (0.9267) (0.9456) (1.1128) (0.9220) (1.0185) (0.8981) (0.9636) (0.8205) (0.8821) (0.7574) (0.8649) (0.7834) (0.8655) (0.7878) (0.9394)

SMB

N

1.9499

∗∗∗

(2)

Lag

−1.5914∗∗−1.8972∗∗−2.2447∗∗−3.2737∗∗∗ −1.4825∗∗−2.1285∗∗−2.0619∗∗−3.8122∗∗∗ −2.0028∗∗−2.5357∗∗−2.3531∗∗∗ −3.9264∗∗∗ −2.5173∗∗∗ −3.1700∗∗∗ −3.0615∗∗∗ −4.3778∗∗∗ −2.9326∗∗∗ −3.7971∗∗∗ −3.6706∗∗∗ −4.7111∗∗∗

0.0018 −0.0187

M.Vola

(1) 1.6479

∗∗∗

4.8278

∗∗∗

(4)

Q2

(0.1390) (0.3719) (0.4941) (1.1417) (0.1263) (0.3654) (0.4447) (1.3748) (0.1164) (0.3353) (0.3768) (1.2537) (0.1184) (0.3147) (0.3396) (1.2018) (0.1200) (0.3109) (0.3074) (1.3090)

∗∗∗

(3)

(2)

RM - RF

CAY

TERMY

DEFY

(Γ − 1)2

(Intercept)

(1)

Q0

Table 2: Predicting GDP growth with (Γ − 1)2 . We regress the quarterly seasonally adjusted GDP growth on the probability distortion index (Γ − 1)2 and control variables DEFY, TERMY, CAY, RM - RF, SMB, HML, MOM, M.Vola, JLN and CATFIN. DEFY is the yield spread between Moody’s BAA and AAA corporate bonds. TERMY denotes the yield spread between long-term government bonds and the T-bill. CAY denotes the detrended wealth. JLN is the measure of economic uncertainty of Jurado et al. (2015). CATFIN is the measure of systemic risk in the financial sector proposed by Allen et al. (2012). Data are obtained from Turan Bali’s, FRED’s, Martin Lettau’s, and Syndey Ludvisgson’s websites. RM - RF, SMB, HML and MOM describe quarterly excess market, size, value and momentum returns, taken from Kenneth French’s data library. M.Vola denotes quarterly stock market volatility. Results are reported in this table. Q0 denotes that no explanatory variable is time-lagged. Q2, Q4, Q6 and Q8 indicate that only (Γ − 1)2 is 2, 4, 6 or 8 quarters lagged, whereas the control variables are always lagged by one quarter. For models (1) to (3) the sample starts in 1953:Q3 and ends in 2015:Q4. Note, that for model (4) the sample starts in 1973:Q1 due to the availability of CATFIN. The symbols *, ** and *** denote the statistical significance at the 10%, 5% and 1% levels. N varies depending the availability of data for the control variables and stands for the number of observations. The R2 is corrected for degrees of freedom. Standard errors are reported in parenthesis and corrected for heteroskedasticity and autocorrelation following Newey and West (1987, 1994).

36

2.0123

1.9078

1.9371

∗∗∗

3.6772

∗∗∗

(4)

0.0802

0.1145

0.0413

adj. R2

N

JLN.FIN

∗∗

1.3298

∗∗∗

(3) 0.2490

(4)

0.1143

2.0226

(1)

(3) 1.5790

∗∗∗

Q4

1.6490

∗∗∗

(2) 0.3126

∗∗∗

(4)

(2)

(3) 1.6958

∗∗∗

Q6

1.8223

∗∗∗

2.0789

(1)

0.5103 ∗∗∗

(4)

2.1125

(1)

(3) 1.7119 ∗∗∗

Q8

1.9224 ∗∗∗

(2)

(4) 0.6758 ∗∗∗

0.2187 −2.7094 −2.4069 −0.9518 −1.4788 −2.0748 −3.4686∗∗−2.4154 −2.8350 −3.9156∗∗−4.1052∗∗∗ −3.4634∗∗−3.4901∗−3.6704∗∗

125

0.0160

0.0034

0.2026

0.0980 −1.1101∗∗∗

0.1841

0.4552∗∗∗

96

125

0.0023 −0.0486 −0.0009

125

(1.3765)

−0.9491

(2.5612)

−0.7727

(0.0744) (0.1189)

−0.0197 −0.0183

(0.0177) (0.0205)

0.0125

(0.0251) (0.0256)

0.0606

0.0125 0.0049

0.0906

125

0.0690

96

0.0163

125

(1.1968)

−1.8709

(2.1540)

6.7989∗∗∗

(0.0728) (0.0976)

0.0173

(0.0264) (0.0243)

0.0233

(0.0301) (0.0203)

(0.0242) (0.0221)

(0.0275) (0.0265) 0.0428 −0.0104

0.0408∗ 0.0259

125

0.0009

(0.0159) (0.0138)

0.0138

(0.0775) (0.0789) (0.0681)

−0.1396∗ −0.1407∗−0.0076

(0.1267) (0.1291) (0.1338)

0.2470∗

(0.3134) (0.3564) (0.4020)

0.0223

0.0345

(0.0133) (0.0174)

−0.0153 −0.0195

(0.0884) (0.0868) (0.0879)

−0.0645 −0.0737

0.0072 −0.0531

(0.1684) (0.1741) (0.2095)

−0.0298

0.0158

125

0.0268

(0.3654) (0.4154) (0.6384)

MOM

JLN

1.5186

∗∗∗

(2)

Begin of sample in each model until 1984:Q3

0.4164∗∗∗

0.0633

125

0.0174 0.0315

0.0971

125

0.0609

96

0.0423

125

(1.3188)

−1.6455

(1.9192)

5.8885∗∗∗

(0.0613) (0.0907)

0.0085

(0.0269) (0.0250)

0.0256

(0.0305) (0.0210)

0.0422 −0.0077

(0.0250) (0.0232)

0.0423∗ 0.0316

(0.0156) (0.0142)

0.0154 −0.0007

(0.0839) (0.0854) (0.0756)

−0.1299 −0.1263 −0.0042

(0.1211) (0.1224) (0.1286)

0.2490∗∗ 0.1825

0.1277 −0.0048 −1.0022∗∗

(0.3149) (0.3671) (0.3917)

0.4097∗∗∗ 0.0252

0.0792

125

0.0049

0.0205 0.0300

0.1166

125

0.1127

96

0.0623

125

(1.3118)

−1.3535

(2.0727)

5.5481∗∗∗

(0.0590) (0.0826)

0.0170

(0.0267) (0.0261)

0.0263

(0.0299) (0.0226)

0.0415 −0.0055

(0.0244) (0.0238)

0.0425∗ 0.0307

(0.0165) (0.0160)

0.0169

(0.0816) (0.0782) (0.0628)

−0.1191 −0.1154

(0.1155) (0.1213) (0.1190)

0.2441∗∗ 0.1727

0.0343 −0.1116 −1.1161∗∗∗ (0.3078) (0.3549) (0.3769)

0.3888∗∗∗

0.1030

125

0.0003

0.0152 0.0555

0.1343

125

(1.2244) 0.1133

96

−1.7475

(2.0815)

5.5351∗∗∗

(0.0604) (0.0800)

0.0268

(0.0265) (0.0248)

0.0229

0.0368 −0.0100

(0.0294) (0.0216)

(0.0243) (0.0239)

0.0431∗ 0.0298

(0.0163) (0.0141)

0.0139

(0.0693) (0.0695) (0.0641)

−0.1254∗ −0.1231∗ 0.0070

(0.1129) (0.1145) (0.1137)

0.2381∗∗ 0.1773

(0.2843) (0.3571) (0.3656)

−0.0140 −0.1545 −1.0930∗∗∗

(1.6773) (1.8504) (1.9667) (1.7313) (1.4248) (1.6545) (1.6418) (1.9056) (2.3160) (2.5875) (2.2102) (2.2441) (1.6637) (2.2073) (1.9478) (1.5708) (1.3959) (1.5585) (1.8559) (1.4352)

−2.3865 −1.8322 −1.5156 −2.7426 −1.3052

0.0267 −0.0146

M.Vola

1.9653

∗∗∗

(1)

Q2

(0.1719) (0.4217) (0.6072) (1.1825) (0.1666) (0.4071) (0.6272) (0.8907) (0.1735) (0.4300) (0.5156) (0.8941) (0.1661) (0.4166) (0.4505) (0.9292) (0.1480) (0.3501) (0.4144) (0.9287)

∗∗∗

(3)

(2)

HML

SMB

RM - RF

CAY

TERMY

DEFY

(Γ − 1)2

(Intercept)

(1)

Q0

Table 3: The ability of (Γ − 1)2 to predict future GDP growth for (1) the beginning of the sample in each model until 1984:Q3 and (2) the subsample 1984:Q4 until 2015:Q4. We regress the quarterly seasonally adjusted GDP growth on the probability distortion index (Γ − 1)2 and control variables DEFY, TERMY, CAY, RM - RF, SMB, HML, MOM, M.Vola, JLN and JLN.FIN (for the first subsample) or CATFIN (for the second subsample). We opt to use JLN.FIN instead of CATFIN for the first subsample to increase the number of observations in model (4) because JLN.FIN starts in 1960:Q3 while CATFIN starts in 1973:Q1, and they are highly correlated with 54%. For models (1)–(3) our sample starts in 1953:Q3 for the first subsample. DEFY is the yield spread between Moody’s BAA and AAA corporate bonds. TERMY denotes the yield spread between long-term government bonds and the T-bill. CAY denotes the detrended wealth. JLN is the aggregate macro uncertainty measure and JLN.FIN the aggregate financial uncertainty measure of Jurado et al. (2015). CATFIN denotes the measure of systemic risk in the financial sector proposed by Allen et al. (2012). Data are obtained from Turan Bali’s, FRED’s, Martin Lettau’s, and Syndey Ludvisgson’s websites. RM - RF, SMB, HML and MOM describe quarterly excess market, size, value and momentum returns, taken from Kenneth French’s data library. M.Vola denotes quarterly stock market volatility. Results are reported in this table. Q0 denotes that no explanatory variable is time-lagged. Q2, Q4, Q6 and Q8 indicate that only (Γ − 1)2 is 2, 4, 6 or 8 quarters lagged, whereas the control variables are always lagged by one quarter. All regressions are done for (1) the beginning of each sample in each model until 1984:Q3 and (2) the subsample 1984:Q4 until 2015:Q4. The symbols *, ** and *** denote the statistical significance at the 10%, 5% and 1% levels. N varies depending the availability of data for the control variables and stands for the number of observations. The R2 is corrected for degrees of freedom. Standard errors are reported in parenthesis and corrected for heteroskedasticity and autocorrelation following Newey and West (1987, 1994).

37

1.2543

2.0590

2.1305

∗∗∗

4.0924

∗∗∗

(4)

125

adj. R2

N

CATFIN

JLN

−0.0001

0.2965

125

0.0037

SMB

0.0204

0.2866

125 0.3712

0.0123

125

(0.8585)

125

−1.2690

(1.4075)

−3.3742∗∗

(0.0521) (0.0347)

0.0091

(0.0053) (0.0060)

−0.0004 −0.0020

(0.0095) (0.0086)

−0.0074 −0.0117

(0.0110) (0.0104)

0.0081

0.0076 −0.0064

(0.0083) (0.0089)

RM - RF

(0.0379) (0.0414) (0.0393)

0.0879∗∗ 0.0363

(0.0574) (0.0605) (0.0651)

(0.3127) (0.3168) (0.3160)

−0.9293∗∗∗ −0.9425∗∗∗ −0.3765

0.0749∗

M.Vola

4.2325

∗∗∗

(4) 1.2878

∗∗∗

(1)

(3) 1.8811

∗∗∗

Q4

1.9514

∗∗∗

(2) 4.0814

∗∗∗

(4) 1.2963

∗∗∗

(1)

(3) 1.9388

∗∗∗

Q6

1.9639

∗∗∗

(2) 3.9929

∗∗∗

(4) 1.3058

∗∗∗

(1)

(3) 2.0058 ∗∗∗

Q8

2.0067

∗∗∗

(2)

(4) 3.8642∗∗∗ ∗∗∗

0.0363 −0.0059

0.0565

0.0072

0.1939

125

0.0080

0.0344

0.2046

125 0.3333

0.0271

125

(0.7150)

125

−1.5060∗∗

(1.8122)

−4.0301∗∗

(0.0459) (0.0304)

0.0140

(0.0045) (0.0044)

−0.0058 −0.0071

(0.0114) (0.0088)

−0.0059 −0.0105

(0.0150) (0.0097)

0.0023

0.0123∗ −0.0041

(0.0068) (0.0068)

(0.0381) (0.0414) (0.0436)

0.0393

(0.0547) (0.0583) (0.0482)

0.0594

(0.2759) (0.2924) (0.3255)

−0.7959∗∗∗ −0.8273∗∗∗ −0.1351 0.0254 −0.0292

0.0561

0.0154

0.1935

125

0.0065

0.0408

0.2072

125

0.3357

0.0344

125

(0.7116)

125

−1.6420∗∗

(1.7861)

−3.8084∗∗

(0.0424) (0.0295)

0.0142

(0.0042) (0.0043)

−0.0063 −0.0072∗

(0.0113) (0.0090)

−0.0066 −0.0120

(0.0149) (0.0096)

0.0020

0.0122∗ −0.0037

(0.0068) (0.0066)

(0.0328) (0.0359) (0.0385)

0.0439

(0.0494) (0.0502) (0.0473)

0.0403

(0.2341) (0.2779) (0.3120)

−0.7526∗∗∗ −0.8066∗∗∗ −0.1199 0.0119 −0.0502

0.0488

0.0167

0.1953

125

0.0061

0.0406

0.2133

125

0.3358

0.0441

125

(0.6859)

125

−1.6227∗∗

(1.7876)

−3.6197∗∗

(0.0455) (0.0337)

0.0098

(0.0043) (0.0043)

−0.0078∗ −0.0084∗

(0.0106) (0.0083)

−0.0064 −0.0108

(0.0147) (0.0095)

0.0017

0.0123∗ −0.0023

(0.0074) (0.0067)

(0.0325) (0.0372) (0.0358)

0.0417

(0.0476) (0.0455) (0.0497)

0.0241

(0.2315) (0.2833) (0.3056)

−0.7365∗∗∗ −0.7863∗∗∗ −0.1288

0.0416

0.0201

0.2070

125

0.0056

0.0399

0.2258

125

0.3406

125

(0.7292)

(1.8912)

−1.6580∗∗

−3.3376∗

(0.0458) (0.0355)

0.0043

(0.0043) (0.0043)

(0.0111) (0.0093) −0.0075∗ −0.0069

−0.0061 −0.0103

(0.0148) (0.0092)

0.0017

0.0128∗ −0.0009

(0.0073) (0.0069)

(0.0309) (0.0326) (0.0371)

0.0354

0.0069 −0.0046 −0.0632

(0.0505) (0.0463) (0.0617)

(0.2411) (0.2877) (0.3453)

−0.7323∗∗∗ −0.7606∗∗∗ −0.1514

(0.5969) (0.5375) (0.7157) (0.6734) (0.4678) (0.6884) (0.7662) (0.7049) (0.5390) (0.6240) (0.6449) (0.5885) (0.5150) (0.6210) (0.7736) (0.5851) (0.5180) (0.6168) (0.6614) (0.7097)

CAY

MOM

(3) 1.8833

∗∗∗

1.9770

∗∗∗

(2)

1984:Q4 – 2015:Q4

−0.5026 −0.4098 −0.1602 −1.0768 −0.8215∗ −1.0547 −0.6671 −1.4609∗∗−1.1571∗∗−1.0206 −0.7547 −1.4386∗∗−1.4056∗∗∗ −1.1739∗ −1.0870 −1.5581∗∗∗ −1.7010∗∗∗ −1.5889∗∗−1.5668∗∗−1.7755∗∗

0.0098 −0.0057 −0.0253

HML

1.2685

∗∗∗

(1)

Q2

(0.1433) (0.3053) (0.3916) (0.7132) (0.1144) (0.2742) (0.3350) (1.0008) (0.0945) (0.2287) (0.2527) (0.9444) (0.1012) (0.2186) (0.2408) (0.9209) (0.1030) (0.2388) (0.2359) (1.0283)

∗∗∗

(3)

(2)

TERMY

DEFY

(Γ − 1)2

(Intercept)

(1)

Q0

Table 3 – Continued

Table 4: Out-of-sample predictability of (Γ − 1)2 . This table reports the results of the out-of-sample 2 analysis. ROS = 1 − M SE1 /M SE0 is the out-of-sample R2 (Campbell and Thompson, 2008) which measures the proportional reduction in mean squared error (MSE) for the predictive regression forecast based on (Γ − 1)2 (M SE1 ) relative to the benchmark forecast (M SE0 ) given in the respective row. The benchmark models are an AR(1) forecast, the random-walk hypothesis and the historical average. While Panel A reports the results of a rolling regression with in-sample size of Nin = 120, Panel B presents the results of a recursive regression. The respective subsequent second rows report the corresponding p-values for the standard Clark 2 and West (2007) MSE-adjusted statistic for testing the null hypothesis that ROS is smaller than or equal 2 to zero against the alternative that ROS greater than zero. The respective subsequent third rows show the p-values that are additionally corrected for heteroskedasticity and autocorrelation according to Newey and West (1987) where the lag choice equals the respective forecast horizon. Q2, Q4, Q6 and Q8 indicate a 2-quarter, 4-quarter, 6-quarter and 8-quarter forecast. Nout denotes the out-of-sample size. The in-sample period starts from 1944:Q2. The prediction period starts from 1977:Q4 and ends 2015:Q4. The symbols *, ** and *** denote the statistical significance at the 10%, 5% and 1% levels applying the Newey-West correction.

ROS in % Forecast horizon

Q2

Q4

Q6

Q8

1.96∗∗

2.64∗

3.45∗

0.262

0.017

0.012

0.004

0.251

0.047

0.059

0.054

24.68∗∗∗

29.33∗∗∗

38.44∗∗∗

43.28∗∗∗

p-value

0.000

0.001

0.000

0.000

p-value (Newey-West)

0.000

0.007

0.002

0.000

18.29∗∗∗

13.03∗∗∗

12.43∗∗∗

9.65∗∗

p-value

0.000

0.000

0.000

0.000

p-value (Newey-West)

0.001

0.004

0.006

0.014

AR(1)

1.90∗∗

4.31∗∗

4.46∗

3.51∗

p-value

0.021

0.001

0.002

0.003

p-value (Newey-West)

0.027

0.041

0.080

0.086

26.78∗∗∗

21.95∗∗∗

33.67∗∗∗

43.42∗∗∗

p-value

0.000

0.000

0.000

0.000

p-value (Newey-West)

0.000

0.006

0.002

0.001

21.74∗∗∗

4.49∗

4.96∗

7.70∗∗

p-value

0.000

0.002

0.001

0.000

p-value (Newey-West)

0.000

0.052

0.079

0.038

A. Rolling regressions, Nin = 120 AR(1) p-value p-value (Newey-West) Random Walk

Historical Mean

−0.27

B. Recursive regressions

Random Walk

Historical Mean

Prediction Period Nout

1977:Q4–2015:Q4 1978:Q2–2015:Q4 1978:Q4–2015:Q4 1979:Q2–2015:Q4 153

151

38

149

147

Table A.1: Selected preference parameter constellations and the equity premium. We set σM = 0.15 to calculate the equity premium.

Parameter

Sharpe Ratio

Equity Premium

α

γ

λ

µeM

0.88

0.65

2.25

µeM σM

0.5

0.075

1

0.65

2.25

0.48

0.072

0.88

1

2.25

0.335

0.05

1

1

2.25

0.323

0.048

α

γ

1

0

0

39