2.1. A parametric example

The aggregator $ f(c,J)$ encodes tradeoffs between current consumption and the future utility index. To better understand how sentiment about the future as well as short-run and long-run risk affect the expected growth rate of consumption, we consider the aggregator of a parametric recursive utility for which existence and unicity are discussed by Schroder and Skiadas (Reference Schroder and Skiadas1999). The time-additve expected utility is a special case of the Schroder and Skiadas (Reference Schroder and Skiadas1999) recursive utility.

2.1.1. Aggregator of the Schroder and Skiadas (Reference Schroder and Skiadas1999)

To estimate equation (6), we rely on the aggregator of the Schroder and Skiadas (Reference Schroder and Skiadas1999)’s parametric recursive utility given by

(9) \begin{equation} f(c,J)=(1+\alpha )\Big [\frac{c^{\gamma }}{\gamma }J^{\frac{\alpha }{1+\alpha }}-\beta J\Big ], \end{equation}

with $\alpha \gt -1$ and $\gamma \lt \min{\left ( 1,\frac{1}{1+\alpha }\right ) }$ . This parametric recursive utility is more general than the time-additive expected utility. Additionally, it simplifies the exposition of the econometric model and the interpretation of the parameters to be estimated.

The discount parameter $\beta$ is assumed to be positive. In a time-additive utility model, $\beta$ would simplify the rate of time preference. Time preference is generally endogenous in nonadditive recursive utility settings. The ratio $\frac{1}{1-\gamma }$ is the elasticity of intertemporal substitution and the parameter $\alpha$ captures the dependency of current utility to future utility $J$ .Footnote ⁷ The curvature of the aggregator $f(x,J)$ with respect to the future utility argument, $J,$ characterizes preferences for the timing of resolution of uncertainty (Kreps and Porteus (Reference Kreps and Porteus1978); Duffie and Epstein (Reference Duffie and Epstein1992); Schroder and Skiadas (Reference Schroder and Skiadas1999); Strzalecki (Reference Strzalecki2013); Epstein et al. (Reference Epstein, Farhi and Strzalecki2014); Zhao (Reference Zhao2017)), with a convex aggregator favoring early resolution, and a concave aggregator favoring late resolution (Skiadas (Reference Skiadas1998)). An additive temporal aggregator then corresponds to indifference towards the timing of resolution. The sign of the product term $\gamma \alpha$ expresses the curvature of the aggregator with respect to the second argument, and it, therefore, captures attitudes towards the temporal resolution of uncertainty.Footnote ⁸ As will be later evidenced in subsections 2.1.3 and 3.1, the parameter $\alpha$ calls attention to the role played by sentiment about the future as well as long-run risks in shaping the dynamic of optimal consumption behavior. Additionally, the sign of the product parameter $\alpha \gamma$ helps understand consumers’ attitudes towards the temporal resolution of uncertainty. A value of $\gamma \alpha$ different from zero expresses nonindifference towards the temporal resolution of uncertainty. A negative sign, $\gamma \alpha \lt 0,$ expresses a preference for early resolution of uncertainty whereas a positive sign, $\gamma \alpha \gt 0,$ expresses a preference for late resolution of uncertainty. In the very particular case where $\alpha =0,$ which corresponds to the indifference towards the timing of resolution, the aggregator of the standard time-additive expected utility is obtained as $f(x,V)=\frac{x^{\gamma }}{\gamma }-\beta J.$

Let us mention that in a deterministic setting, a world without uncertainty $(\sigma =0),$ concerns for temporal resolution of uncertainty is not relevant ( $\alpha =0$ ), and the Schroder and Skiadas (Reference Schroder and Skiadas1999) parametric recursive utility reduces to the aggregator of the time additive expected utility, that is $f(x,J)=\frac{x^{\gamma }}{\gamma }-\beta J.$

2.1.2. Production function

Let us assume the per capital production function is given by

(10) \begin{equation} F(K(t))=A(t)(K(t))^{\nu }, \end{equation}

where $0\lt \nu \leq 1$ and parameter $A(t)\gt 0$ is the total factor productivity (TFP) at time $t$ . This functional form includes the A(t)K(t) production function as a special case, $(\nu =1)$ .

2.1.3. Optimal consumption path

Substituting the production function (10) and the parametric aggregator specification (9) in the expected consumption growth (6) gives

(11) \begin{eqnarray} \frac{1}{c}\frac{1}{dt}E_{t}dc &=& \frac{1}{1-\gamma }\left [\left (-\beta (1+\alpha )+\alpha \frac{c^{\gamma }}{\gamma }J^{\frac{-1}{1+\alpha }}\right )+(\nu AK^{\nu -1}-\delta ) +\underbrace{\left [\frac{\alpha }{1+\alpha }\right ]\left (\frac{1}{J}\frac{1}{dt}E_{t}dJ\right )}_{\text{Sentiment}}\right ] \notag \\ &+& \frac{1}{2}\left [\underbrace{(2-\gamma )\sigma _{c}^{2}}_{\text{Short-run risk}}+\underbrace{\frac{\alpha }{(1+\alpha )^{2}(\gamma -1)}\sigma _{J}^{2}(t)-\frac{ 2\alpha }{1+\alpha }\sigma _{cJ}(t)}_{\text{Long-run risk}}\right ] \end{eqnarray}

It is also worth mentioning that the special case where $\alpha =0$ which relates to a time-additive expected utility, expresses indifference towards the resolution of uncertainty. In that case, equation (11) reduces to

(12) \begin{eqnarray} \frac{1}{c}\frac{1}{dt}E_{t}dc &=& \frac{1}{1-\gamma }\left [-\beta +(\nu AK^{\nu -1}-\delta )+\frac{1}{2}\underbrace{(2-\gamma )\sigma _{c}^{2}}_{\text{Short-run risk}} \right ] \end{eqnarray}

As we can see in equation (12), the special case where $\alpha =0$ would imply that consumption variations are not impacted neither by sentiment component accounts nor by long-run risks. The statistical significance of the preference parameters in estimating the consumption path (11) will be of paramount interest in the empirical section.

3.1. Estimation of the parameters

The estimation of the parameters $\alpha$ and $\gamma$ requires to obtain the proxies of variables in both the right-hand side of the optimal consumption rule (11). The left-hand side corresponds to the optimal consumption. It will be estimated using the method of estimating continuous time diffusion process of Nowman (Reference Nowman1997). On the other hand, the variables that appear on the right-hand side of the optimal consumption (11) include the value function and its growth rate. While the value function is not directly observable, it depends on expectations about future prospects of the economy. To emphasize this point, Koopmans (Reference Koopmans1960) refers to the value function as the prospective utility. The value function can be estimated by using a small numbers of factors $F=(F_{1},\ldots,F_{L})$ called latent variables that capture expectations about future prospects of the economy. The estimation of latent factors is based on the procedure proposed by Bai and Ng (Reference Bai and Ng2002) and presented in Appendix A. Other studies that have used the latent factor approach include Bai and Ng (Reference Bai and Ng2002); Bernanke and Boivin (Reference Bernanke and Boivin2003); Bernanke et al. (Reference Bernanke, Boivin and Eliasz2005); Favero et al. (Reference Favero, Marcellino and Neglia2005); Boivin and Giannoni (Reference Boivin and Giannoni2006); Forni et al. (Reference Forni, Giannone, Lippi and Reichlin2009); Ludvigson and Ng (Reference Ludvigson and Ng2009); Bouaddi and Taamouti (Reference Bouaddi and Taamouti2012); Bouaddi and Taamouti (Reference Bouaddi and Taamouti2013); Kakeu and Bouaddi (Reference Kakeu and Bouaddi2017).

In general, there is no closed-form solution to the value function. An approximation of the value function is used for empirical purposes. The empirical estimate of models with a recursive utility framework is still difficult due to the latency of the value function (Chen et al. (Reference Chen, Favilukis and Ludvigson2013); Thimme (Reference Thimme2017)), and researchers must use approximations. While the utility index itself is not observable, one way to deal with the estimation is to substitute it as a function of quantifiable variables, such as latent factors, related to the state of the economy (Cochrane (Reference Cochrane2017)). In a recursive utility framework, the value function depends upon expectations about future consumption growth. Proxying the recursive utility index by using latent factor techniques that capture expectations about the future of the economy is well grounded following tradition in empirical works using recursive utility (Cochrane (Reference Cochrane2005); Hansen (Reference Hansen2010, Reference Hansen2012); Chen et al. (Reference Chen, Favilukis and Ludvigson2013); Kakeu and Bouaddi (Reference Kakeu and Bouaddi2017)). For instance, Chen et al. (Reference Chen, Favilukis and Ludvigson2013) use latent factor techniques to explicitly estimate the unobservable continuation value of the future consumption plan in a discrete time Epstein and Zin (Reference Epstein and Zin1989) recursive utility framework. We use a Schroder and Skiadas (Reference Schroder and Skiadas1999) continuous-time stochastic recursive preferences, and we assume that the logarithm of the value function is a linear function of the estimated latent factors related to sentiment about the future.Footnote ⁹ We will use the Michigan consumer sentiment index which is a monthly assessment of consumer expectations about the future. The Michigan survey is a leading indicator that attempts to predict economic conditions a full year into the future.

Our approach uses a latent factor approximation of the value function which takes the following functional form.

(13) \begin{equation} J(t)=e^{\left ( \theta +\sum _{i=1}^{L}\phi _{i}F_{i}(t)\right ) }. \end{equation}

Equation (13) can be viewed as a way of log linearizing the model. Note that the differential form of equation (13) implies that changes in the value function can be approximated linearly by changes in the latent factors. Log-linear approximations are often used in macroeconomic and financial models (Bansal and Yaron (Reference Bansal and Yaron2004); Ljungqvist and Sargent (Reference Ljungqvist and Sargent2004); Cochrane (Reference Cochrane2005); Lau and Ng (Reference Lau and Ng2007); Restoy and Weil (Reference Restoy and Weil2011)). The estimation of the latent factors is done in the first step of the econometric procedure. The approximated value function $\widetilde{J}(t)=J(\widetilde{F_{1}}_{t},\ldots,\widetilde{F_{L}}_{t})$ is a function of those estimated latent factors obtained from sentiment variables. The estimated value function $\widetilde{J}(t)$ is a consistent estimate of the value function $J(t)$ that is not observable.

When the value function is replaced by its approximate in the pricing equation, the estimation of the parameters $\theta$ , and $(\phi _{i})_{i=1,..L}$ in the expression (13) are done simultaneously with the preference parameters $\alpha$ , $\beta$ , and $\gamma$ described in the expression (9). That is, this specification allows the value function to be endogenously and jointly estimated with the preference parameters in the structural model.Footnote ¹⁰

The optimal consumption rule (11) reduces to

(14) \begin{align} \nonumber \frac{1}{c}\frac{1}{dt}E_{t}dc &= \frac{1}{1-\gamma }\left[\left ( -\beta (1+\alpha )+\alpha \frac{c^{\gamma }}{\gamma }\left ( e^{\left ( \theta +\sum _{i=1}^{L}\phi _{i}F_{i}(t)\right ) }\right ) ^{\frac{1}{1+\alpha }} \right ) +(\nu AK^{\nu -1}-\delta ) \phantom{\underbrace{\frac{\alpha }{1+\alpha }\left ( \sum _{i=1}^{L}\phi _{i}F_{i}(t)+\theta \right ) }}\right.\\&\quad\quad\qquad \left.+\underbrace{\frac{\alpha }{1+\alpha }\left ( \sum _{i=1}^{L}\phi _{i}F_{i}(t)+\theta \right ) }_{\text{Sentiment}}\right ] \nonumber\\ &\quad + \frac{1}{2}\left \{ \underbrace{(2-\gamma )\sigma _{c}^{2}(t)}_{\text{Short-run risk}}+\underbrace{\frac{\alpha }{(1+\alpha )^{2}(\gamma -1)}\sum _{i=1}^{L}\phi _{i}^{2}F_{i}^{2}(t)\sigma _{F_{i}}^{2}(t)-\frac{2\alpha }{1+\alpha }\sum _{i=1}^{L}\phi _iF_{i}(t)\sigma _{cF_{i}}(t)}_{\text{Long-run risk}}\right \} \end{align}

The factor decomposition provided by equation (14) gives a description of the relationship between the expected optimal consumption, sentiment, and long-run risks. Following Hansen (Reference Hansen2010, Reference Hansen2012), this shows that recursive preferences provide a channel for sentiment to matter in consumption decision-making. The present work aims at quantifying the impact of sentiments and long-run risks on optimal consumption decisions. The long-run risk components in equation (14) encapsulate the uncertainty shocks to the latent factors, which follow stochastic processes. Movements in the latent factors should be traced to movements in sentiment indicators. The sentiment indicators are forward-looking indicators as they capture information about the state of the economy as well as expectations about future prospects of the economy.

3.2. Description of the data on consumption per capita

Consumption data come from the St. Louis Federal Reserve Bank and cover the period from February 1980 to December 2014. We used the real personal consumption expenditures of services, the real personal consumption expenditures of nondurable goods, and capital stock at constant national prices.Footnote ¹¹ We also use the total civilian population to get the real consumption per capita and the real capital stock per capita.Footnote ¹² We computed the per capita real consumption as the sum of the real personal consumption expenditures of services and real personal consumption expenditures of nondurable goods over civilian population to get the per capita consumption level. Similarly, we compute the per capita capital as the real capital stock over civilian population. We then use the growth rate (log-difference) on real per capita personal consumption expenditures of services and nondurable goods (See Figure 1).

3.2.1. Estimation of the expected growth rate of consumption $\frac{1}{c}\frac{1}{dt}E_{t}dc$

The consumption is expected to follow a stochastic process

(15) \begin{equation} dC(t) = \mu _{C}(t,C(t))dt+\sigma _{C}(t,C(t))d\zeta (t), \end{equation}

where

\begin{equation*} C(t)=\int _{0}^{t}\frac {dc (\tau )}{c (\tau )} \end{equation*}

represent the cumulative consumption, and $\mu _{C}(t,C(t))=\frac{1}{dt}E_{t}dC=\frac{1}{c}\frac{1}{dt}E_{t}dc$ represents the expected growth rate of consumption.

Figure 1. Consumption per capita growth rate.

For estimation purposes, let us assume the following parametric specification

(16) \begin{align} dC(t)=(\alpha _{1}+\alpha _{2}C(t))dt+\delta C(t)^{\alpha _{3}}d\zeta (t), \end{align}

for which a discrete approximation was developed by Nowman (Reference Nowman1997) and based on some results found in Bergstrom (Reference Bergstrom1983) as follows:

(17) \begin{equation} C_{t}=e^{\alpha _{2}}C_{t-1}+\frac{\alpha _{1}}{\alpha _{2}}(e^{\alpha _{2},}-1)+\eta _{t} \end{equation}

where the conditional distribution of the error term $\eta _{t}$ satisfies

\begin{equation*} E_{t-1}(\eta _{s}\eta ^{t})=\displaystyle \begin {cases} 0 & \mathrm {if\ }s\neq t, \\ \\[-7pt] \frac {\delta ^{2}}{2\alpha _{2}}(e^{\alpha _{2}}-1)(C_{t-1})^{2\alpha _{3}} & \mathrm {if\ }s=t.\end {cases}\end{equation*}

The logarithm of the Gaussian likelihood function is

\begin{equation*} L(\alpha _{1},\alpha _{2},\alpha _{3},\delta )=\sum _{t=1}^{T}[\log E_{t-1}(\eta _{t}^{2})+\frac {(C_{t}-e^{\alpha _{2}}C_{t-1}-\frac {\alpha _{1}}{\alpha _{2}}(e^{\alpha _{2}}-1))^{2}}{E(\eta _{t}^{2})}], \end{equation*}

where $E_{t-1}(\eta _{t}^{2})=\frac{\delta ^{2}}{2\alpha _{2}}(e^{\alpha _{2}}-1)(C_{t-1})^{2\alpha _{3}}.$

Maximum likelihood estimation consists of solving

(18) \begin{equation} (\widehat{\alpha }_{1},\widehat{\alpha }_{2},\widehat{\alpha }_{3},\widehat{\delta })=\arg \max _{\alpha _{1},\alpha _{2},\alpha _{3},\delta }L(\alpha _{1},\alpha _{2},\alpha _{3},\delta ). \end{equation}

It follows from equation (16) that

(19) \begin{equation} \frac{1}{c}\frac{1}{dt}E_{t}dc=\widehat{\alpha }_{1}+\widehat{\alpha }_{2}C(t). \end{equation}

3.3. Description of data on total factor productivity and real capital stock

The optimal consumption path (14) incorporates the marginal product of capital at time $t$ , which depends upon the TFP, $A(t),$ and the capital stock per capita, $K(t)$ .

The capital stock is computed at constant 2005 national prices (in mil. 2005 US $\$ $ ). Capital stock is estimated based on accumulated and depreciated past investments. It includes Structures (residential and nonresidential), Transport equipment, Computers, Communication equipment, Software, and Other machinery and assets.Footnote ¹⁴

The TFP is the output less the contribution of capital and labor. The TFP is adjusted for capacity utilization of the capital stock. The computation methodology related to the utilization-adjusted TFP is discussed by Fernald and Matoba (Reference Fernald and Matoba2009); Fernald (Reference Fernald2014), and Basu et al. (Reference Basu, Fernald and Kimball2006).

We compute the monthly equivalent by using an interpolation technique for deriving a monthly series from annual data. A similar frequency conversion technique is used by the Federal Reserve Bank.Footnote ¹⁵ For more details about the disaggregation of low frequency data to higher frequency, we refer the reader to the following papers: Boot et al. (Reference Boot, Feibes and Lisman1967); Denton (Reference Denton1971); Chan (Reference Chan1993); Feijoo et al. (Reference Feijoo, Caro and Quintana2003); and Feijoo et al. (Reference Feijoo, Caro and Quintana2003) among others. In our context, the measurement error resulting from frequency conversion of the dependent variable has zero mean and and is uncorrelated with regressors. Thus, we can estimate consistently the parameters in this case. We argue that this is true in our case since the high frequency counterpart is obtained by pure statistical method not involving the regressors which testify that the measurement error is independent of regressors preserving the consistency of the estimator. Of course, the estimates will be less precise than with data without measurement error. The induced inflation in the variance of the estimator will impact the significance of the parameters via lower t-statistics. However, if all p-values are below the significant level then this variance inflation is unimportant (the t-statistics are very conservative toward the null hypothesis).

3.4. Database on the University of Michigan’s Survey of Consumers’ sentiment

Sentiment factors are derived from the monthly Survey of Consumers by the University of Michigan.Footnote ¹⁶ The survey on consumer expectations focuses on consumer’s view prospects for their own financial situation, their prospects about the general economy over the near term, and their prospects about the economy over the long term. The survey contains consumer’s prospects about personal finances, savings and retirement, economic conditions, unemployment, prices, government expectations, household goods buying conditions, vehicle buying conditions, and home buying and selling conditions.Footnote ¹⁷

3.5. Calibrated parameters

Some parameters are calibrated to match stylized facts. Following a calibration methodology emphasized by Lucas (Reference Lucas1980) and Kydland and Prescott (Reference Kydland and Prescott1982), the parameter of the production function $\nu$ is set at $0.36$ in line with standard economic research. The subjective discount rate $\beta$ is set at 4% per annum, which is equivalent to 0.33% per month. The capital depreciation rate, $\delta$ , is set at $2.54\%$ per annum, which is equivalent to $0.25\%$ per month.

3.6. Empirical results

The estimation of the parameters related to the optimal consumer path uses an econometric approach that incorporates the latent factor analysis on sentiment indicators. We used the information criteria of Bai and Ng (Reference Bai and Ng2002) to select the optimal number of fundamental factors governing the sentiment indicators. The criterion selected one factor, denoted hereafter by $F$ without subscript. More detail about the related econometric model is provided in Appendix B.

(20) \begin{align} \frac{1}{c}\frac{1}{dt}E_{t}dc &=\frac{1}{1-\gamma }\left [\! \left (\! -\beta (1+\alpha )+\alpha \frac{c^{\gamma }}{\gamma }\left ( e^{\left ( \theta + \phi F (t)\right ) }\right ) ^{\frac{1}{1+\alpha }} \right ) +(\nu AK^{\nu -1}-\delta )+\underbrace{\frac{\alpha }{1+\alpha }\left (\phi F(t)+\theta \right ) }_{\text{Sentiment}}\right ] \nonumber \\ &\quad +\frac{1}{2}\left \{ \underbrace{(2-\gamma )\sigma _{c}^{2}(t)}_{\text{Short-run risk}}+\underbrace{\frac{\alpha }{(1+\alpha )^{2}(\gamma -1)}\phi ^{2}F^{2}(t)\sigma _F^{2}(t)-\frac{2\alpha }{1+\alpha }\phi F(t)\sigma _{cF}(t)}_{\text{Long-run risk}},\right \} \end{align}

Estimated coefficients of the structural model are reported in Table 1.

Table 1. Estimation of preference parameters of the model equation (20)

All the parameters are statistically different from zero, including the parameter $\alpha$ related to risk attitudes towards long-run uncertainty. A decomposition of consumption variations incorporates multiple component including sentiment, short-run risk, and long-run risk, as shown in Table 3. To contrast, with a time-additive expected utility, consumption variations are not impacted neither by sentiment component nor by long-run risks.

With a Schroder and Skiadas (Reference Schroder and Skiadas1999) parametric recursive utility, the sign of the product of the parameters $\gamma \alpha$ is important for understanding consumers’ attitudes towards long-run risk associated with future growth prospects. Occurrence of a negative sign for the product of the parameters $\gamma \alpha \lt 0,$ would mean that consumers prefer an early resolution of uncertainty. And therefore, consumers are averse to long-run risks associated with future growth prospects. As shown in Table 7, the product of the estimated parameters $\gamma \alpha$ is negative and the parameters $\alpha$ and $\gamma$ are statistically significant. This suggests that consumers prefer early resolution of uncertainty, and therefore are averse to long-run risk associated with uncertainty shocks to future growth prospects. This underscores the importance of the long-run risk channel for understanding the consumption path. Our results echo Bansal and Yaron (Reference Bansal and Yaron2004) and Sargent (Reference Sargent2007) who emphasize that economic models incorporating long-run risks have the potential to provide additional channel for understanding dynamic consumers’ behavior.

3.7. Descriptive statistics and correlation statistics between sentiment and risk factors

Using the components of the dynamic optimal consumption path (20), we computed the correlations between sentiment, short-run risks, and long-run risks. As shown in Table 2, there is a statistical significant negative correlation between sentiment and long risks.

Table 2. Correlation between sentiment about the future and risk factors

Note: The t-statistics are between brackets.

This suggests that worsening sentiment about future prospects of the economy is linked to growing long-run uncertainty level about the economy. Long-run uncertainty is high when consumers are less confident in the future prospects of the economy. This suggests that ignoring sentiment about future prospects while analyzing long-run risk involved in consumption decisions would not provide the full picture for understanding forces that govern consumption decisions. The statistical significant correlation also suggests a linkage between sentiment about the future and nonindifference towards the temporal resolution of future uncertainty. In Appendix C, Table 10, we have provided additional descriptive statistics related to sentiment and short-run and long-run risk factors.

4.1. Democratic decomposition of Löwdin (Reference Löwdin1970) applied to equation (21)

In a discrete time setting $t=1,\ldots,T$ , the right-hand side components of equation (21) are the columns of the following matrix

(26) \begin{equation} X=\left ( \begin{array}{cccc} x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \\ . &. &. &. \\ x_{1,t} & x_{2,t} & x_{3,t} & x_{4,t} \\ . &. &. &. \\ x_{1,T} & x_{2,T} & x_{3,T} & x_{4,T}\end{array}\right ) \quad \end{equation}

If the columns of the matrix $X$ are statistically independent or orthogonal, then the ratio of the variance of the ith column to the sum of all the variances of all columns would capture the contribution of that factor in explaining variations in expected consumption growth. But in general, the columns of the matrix $X$ may not be statistically independent or orthogonal. In this case, it is possible to find an orthogonal equivalent of $X$ whose columns are statistically independent or orthogonal, with the diagonal entries being exactly equal to the variances of the components of the columns of $X_{t}$ . The democratic decomposition method developed by Löwdin (Reference Löwdin1970) is a statistical method that allows to transform a matrix of correlated variables into an information-equivalent matrix of variables that are noncorrelated. The democratic decomposition allows to isolate the specific contribution of each component, which is important in light of common variation, permitting a clear interpretation of the individual relationships.Footnote ²⁰ Applied to our economic framework, the democratic decomposition extracts standalone orthogonal components of sentiment, short-run risk, and long-run risk while maintaining an optimal relationship with the underlining variables. The variances of orthogonal components of sentiment, short-run risk, and long-run risk emerging from the democratic decomposition are identical to those of the original variables. The democratic decomposition ensures that the orthogonal components of sentiment, short-run risk, and long-run risk best resemble the original variables. Using this procedure, the orthogonal components are used to compute the relative contribution of sentiment, short-run risk, and long-run risk in explaining variations in expected consumption growth.

In what follows we give a brief presentation of the democratic orthogonalization method developed by Löwdin (Reference Löwdin1970). Let us consider the general case where the components of the vector $X_{t}= ( x_{1,t},x_{2,t},x_{3,t},x_{4,t} ) ^{\prime }$ may display some correlation. In order to analyze the contribution of each component, we need to find an orthogonal equivalent, denoted by $Z_{t}= ( z_{1,t},z_{2,t},z_{3,t},z_{4,t} ) ^{\prime }$ whose associated covariance matrix is diagonal and the diagonal entries are exactly the variances of the components of the vector $X_{t}$ .

(27) \begin{equation} \mathcal{X}=\left ( \begin{array}{cccc} x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \\ . &. &. &. \\ x_{1,t} & x_{2,t} & x_{3,t} & x_{4,t} \\ . &. &. &. \\ x_{1,T} & x_{2,T} & x_{3,T} & x_{4,T}\end{array}\right ) -\left ( \begin{array}{cccc} \overline{x}_{1} & \overline{x}_{2} & \overline{x}_{3} & \overline{x}_{4} \\ . &. &. &. \\ \overline{x}_{1} & \overline{x}_{2} & \overline{x}_{3} & \overline{x}_{4} \\ . &. &. &. \\ \overline{x}_{1} & \overline{x}_{2} & \overline{x}_{3} & \overline{x}_{4}\end{array}\right ) \end{equation}

Denote by

\begin{equation*} \widehat {\Omega }=\text{Cov}(\mathcal {X}) \end{equation*}

the sample covariance matrix of the components of $\mathcal{X}$ . It can be factorized as

(28) \begin{align} \widehat{\Omega }=F\Lambda F^{\prime } \end{align}

(29) \begin{align} F^{\prime } F=I \end{align}

(30) \begin{align} \Lambda \,\,\text{is a diagonal matrix}. \end{align}

As shown by Löwdin (Reference Löwdin1970), the democratically orthogonalized components are given by

(31) \begin{equation} \mathcal{Z}=Y\Upsilon. \end{equation}

where

(32) \begin{align} Y=\mathcal{X}F\Lambda ^{-\frac{1}{2}}F^{\prime }, \end{align}

(33) \begin{align} \Upsilon =\text{Diag}(\widehat{\Omega }) \end{align}

To be clear, the matrix $\Upsilon =\text{Diag}(\widehat{\Omega })$ is the diagonal matrix whose main diagonal is equal to the main diagonal of $\widehat{\Omega }$ . Note also that $Y^{\prime }Y=I$ .

The variances of the columns of the matrix (31) resulting from the democratic decomposition

are identical to the variances of the columns of the initial matrix of components $X$ [shown in equation (26)].

To estimate the cross-section variances $(s_{i,t}^{2})_{i=1,2,3,4\,\, t=1.,,,,T}$ related to the entries of $\mathcal{Z}$ , represented by the matrix form as follows

(34) \begin{equation} s^{2}=\left (\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} s_{1,1} ^2 & s_{2,1}^2 & s_{3,1}^2 & s_{4,1}^2 \\ . &. &. &. \\ s_{1,t} ^2 & s_{2,t}^2 & s_{3,t}^2 & s_{4,t}^2 \\ . &. &. &. \\ s_{1,T}^2 & s_{2,T}^2 & s_{3,T}^2 & s_{4,T}^2\end{array}\right ) \end{equation}

we used 200 bootstraped $(\mathcal{Z})_{b=1,\ldots,200}$ samples of $\mathcal{Z}$ . Using this procedure, the estimated cross-section variances of $\mathcal{Z}$ are used to compute the relative contribution of sentiment, short-run risk, and long-run risk in explaining variations in expected consumption growth as follows:

(35) \begin{align} \text{Sentiment impact}=\frac{s_{2,t}^{2}}{s_{1,t}^{2}+s_{2,t}^{2}+s_{3,t}^{2}+s_{4,t}^{2}}\, \, \, \,\,\,\,\,\,\,\,\, \, \, \,\,\,\,\,\,\,\,\,\,\,\,\quad t=1,\ldots,T. \end{align}

(36) \begin{align} \text{Long-run risk impact}=\frac{s_{3,t}^{2}}{s_{1,t}^{2}+s_{2,t}^{2}+s_{3,t}^{2}+s_{4,t}^{2}}\, \, \, \,\,\,\,\,\,\quad t=1,\ldots,T. \end{align}

(37) \begin{align} \text{Short-run risk impact}=\frac{s_{4,t}^{2}}{s_{1,t}^{2}+s_{2,t}^{2}+s_{3,t}^{2}+s_{4,t}^{2}}\, \, \, \,\,\,\,\,\,\quad t=1,\ldots,T. \end{align}

(38) \begin{align} \text{Macro risk impact}=\frac{s_{3,t}^{2}+s_{4,t}^{2}}{s_{1,t}^{2}+s_{2,t}^{2}+s_{3,t}^{2}+s_{4,t}^{2}}\, \, \, \,\,\,\,\,\,\quad \quad \quad t=1,\ldots,T. \end{align}

4.2. Descriptive statistics of impact proportions

Table 3 displays the descriptive statistics of the sentiment impact, the short-run risk, the long-run risk, and the macro-risk impact, which is the sum of the short-run risk and long-run risk impacts. Table 3 shows that on average the sentiment component accounts for 15.33% of variations in consumption while macroeconomic risks component account for 51.39%, of which 16.89% pertains to the short-run risk and 34.51% pertains to the long-run risk.

Table 3. Descriptive statistics of the sentiment impact, the short-run risk impact, the long-run risk impact, and the macro-risk impact

Figure 2 displays the estimations of the contribution of the sentiment variations to changes in the variation of the expected consumption from 1980 to 2014.

Figure 2. Sentiment impacts from 1980 to 2014.

The graphic analysis of Figure 2 shows that after economic crises (darker shaded areas) the sentiment impact experiences increases or bumps. The highest bump in sentiment impact occurs after early 1990s economic crisis in the USA while the highest bump in the macro risk occurs after the early 1980s recession. The sentiment impacts from financial crisis (light shaded areas) appear to be lower than the ones that occur during or after nonfinancial crises (dark-shaded areas). Sentiment contributions appear to be high during NBER recessions and low during financial crises.

Figure 3 displays the estimations of the Macroeconomic risk impacts from 1980 to 2014. The macroeconomic risk impact varies over time but stays more often greater than 48%. Some bumps in the impact of macroeconomic risk are observed after the early 1980s crises, east Asian crisis (1997), stock market downturn of 2002, and during of after 2007–2009 crises 2009.

Figure 3. Macroeconomic risk impacts from 1980 to 2014.

Figure 4 displays the estimations of the long-run risk impacts from 1980 to 2014. The long-run risk is the dominant component of the macroeconomic risk impact in consumption, staying more often above 30%. The long-risk impact follows a similar pattern as the macroeconomic risk. One important fact to be hilighlighted from the graph is that the long-run risk impacts appear to be leading indicators of NBER recessions and financial crises. That is, the long-run impacts start increasing a few years before crises. Bumps in the long-run risk impacts are observed during the following periods: early 1998s crisis, 1991–1992, financial crash of 1987, the east Asian crisis (1997)

Figure 4 Long-run risk impacts from 1980 to 2014.

Figure 5 below displays the estimations of the short-run risk impacts from 1980 to 2014. The short-run risk impact represents on average 15% of changes in consumption. The short-run risk impact displays bumps after most of the crises.

Figure 5. Short-run risk impacts from 1980 to 2014.

Figures 4 and 5 show that the long-run risk proportion is higher at the start of recession periods while the short-run risk proportion decreases at the start of recession periods. This fact is related to the consumer behavior, as consumer tends to be more pessimistic at the beginning of the recession periods and becomes more worried about the long-run impact of the crisis driving him/her to increase heavily his/her precautionary saving and reduce his/her consumption. At the end of the crisis, the conditions become more stable and the consumer becomes more optimistic and the short-run risk becomes more of a concern leading to a decrease in the long-run risk and an increase of the importance of the short-run risk.

4.3. Comparison with a time-additive expected utility model

This section provides an empirical comparison of the recursive utility model with sentiments and that of the time-additive utility without sentiments. Say another way, we address the question of how well the recursive utility model explains data relative to competing for time-additive utility specification. To rank the two competing models, we use the statistical criterion proposed by Akaike (Reference Akaike1974), also known as the Akaike information criterion (AIC). The AIC is grounded in information theory. It quantifies the information loss when the true model of the data is not selected.

With a time additive expected utility, which corresponds to the aggregator $f(c,J)=U(c)-\beta J$ , where $U(c)=\frac{c^{\gamma }}{\gamma }$ , (which corresponds to the case $\alpha =0$ ), the problem of the representative agent is written as

(39) \begin{equation} \max _{\{c(t):\,t\geq 0\}}E_{0}\Big [\int _{0}^{\infty }e^{-\beta t}U(c(t))dt\Big ], \end{equation}

subject to:

(40) \begin{align} dK(t) =\Big [F(K(t))-c(t)-\delta K(t)\Big ]dt+\sigma (K(t))dB(t), \end{align}

(41) \begin{align} c(t)\geq 0, \end{align}

(42) \begin{align} K(t)\geq 0, \end{align}

(43) \begin{align} K(0)=K_{0}\gt 0. \end{align}

and the optimal consumption path should follow the following rule

(44) \begin{equation} \frac{1}{c(t)}\frac{1}{dt}E_{t}dc(t)=\left ( \frac{-c(t)u_{cc}(t)}{u_{c}(t)}\right ) ^{-1}[(F_{K}(t)-\delta )-\beta ]-\frac{1}{2}\left [ -\frac{c^{2}(t)u_{ccc}(t)}{c(t)u_{cc}(t)}\right ] \sigma _{c}^{2}(t), \end{equation}

which reduces to

(45) \begin{equation} \frac{1}{c(t)}\frac{1}{dt}E_{t}dc(t)=\left ( 1-\gamma \right ) ^{-1}[F_{K}(t)-\delta -\beta ]-\frac{1}{2}\left [ 1-\gamma \right ] \sigma _{c}^{2}(t), \end{equation}

The parameter estimates and statistical tests of the time-additive model are reported in Table 4. For the time-additive utility and recursive models, the parameter of the production function $\nu$ is set at $0.36$ in line with standard economic research. The subjective discount rate $\beta$ is set at 4% per annum, which is equivalent to 0.33% per month. The capital depreciation rate, $\delta$ , is set at $2.54\%$ per annum, which is equivalent to $0.25\%$ per month. As shown in Table 4, the time-additive expected utility model yields an implausibly large value of $1-\gamma =4.09$ for the risk-aversion parameter. The latter issue is related to the so-called Equity Premium Puzzle, a term coined by Mehra and Prescott (Reference Mehra and Prescott1985) to describe the improbably high-risk aversion one must have, in the context of standard time additive expected utility, to own risk-free bonds given the immense equity return premium offered by equity markets.

Table 4. Parameters estimation with the standard time-additive expected utility model

Table 5 reports the measure of specification error given by the AIC information criterion for the recursive utility and the time-additive utility models discussed above. The AIC information criterion can be used to establish the quality of a statistical model for a given set of data (Akaike (Reference Akaike1974)). The AIC information criterion then allows models to be compared, with the having the lowest value being preferable for a given set of data.

Table 5. Akaike (1974) information criterion (AIC) for recursive utility and time-additive utiliy

We can see that the estimated recursive utility model always displays a smaller AIC value than the time-separable CRRA model. The AIC value for the recursive utility specification is $8.6E+06$ , about 94% smaller than that of the time-separable CRRA model.

4.4. Robustness

In this section, we perform a robustness check (sensitivity analysis) by estimating all the parameters except the depreciation rate of capital which is calibrated to different values. The reason behind the calibration of the depreciation rate of capital instead of estimating it is that the discount rate and the depreciation rate of capital are not simultaneously identifiable as can be easily seen from equation (20). Therefore, we choose to calibrate the depreciation rate of capital. We followed the literature to calibrate this rate. This includes Kydland and Prescott (Reference Kydland and Prescott1982) and Greenwood et al. (Reference Greenwood, Hercowitz and Huffman1988) who used a depreciation rate of 0.1 while Dejong et al. (Reference Dejong, Ingram and Whiteman2000) used a depreciation rate ranging from 0.03 and 0.17 and Gomme and Rupert (Reference Gomme and Rupert2007) used a rate of 0.0391. For the sake of robustness in our analysis, we used the minimum and the maximum of the above rates as well as the median. Hence, the depreciation rate is set to 0.03, 0.17, and 0.1 per annum. Tables 6–8 below give the estimation of the other parameters for the three values of the depreciation rate. We notice from the three tables (Tables 6–8) that the estimates of the other parameters and their significance are barely sensitive to the depreciation rate changes.

4.4.1. Tests of the difference between the consumption growth rate implied by equation (20) (recursive utility) and the one implied by equation (45)(standard time-additive utility)

To test whether the model with recursive utility is still quantitatively different from the one with time additive utility even if the parameter $\alpha$ looks small in the aggregator, we computed the consumption growth rate implied by equation (20) (recursive utility) and the one implied by equation (45) (time additive utility) and conducted the t-test for the equality of the means, the Wilcoxon/Mann–Whitney test for the equality of the medians and the F-test for equality of variances.

Table 9 shows that the three tests strongly reject the null hypothesis as the p-values are below any conventional significance level.

Table 6. Estimation of preference parameters of the model equation (20) when $\delta =0.17$

Table 7. Estimation of preference parameters of the model equation (20) when $\delta =0.1$

Table 8. Estimation of preference parameters of the model equation (20) when $\delta =0.03$

Table 9. Tests of the difference between the consumption growth rate implied by equation (20) (recursive utility) and the one implied by equation (45) (time additive utility), when $\delta =0.03$

This result is enforced by the kernel nonparametric density estimation (Figure 6) of the difference between the consumption growth rate implied by equation (20) (recursive utility) and the one implied by equation (45) (time additive utility).

Figure 6. Density distribution of the difference between the consumption growth rate implied by equation (20) (recursive utility) and the one implied by equation (45) (time additive utility), when $\delta =0.03$ . For two other alternative values, see Figures 7 and 8.

Figure 7. Density distribution of the difference between the consumption growth rate implied by equation (20) (recursive utility) and the one implied by equation (45) (time additive utility), when $\delta =0.1$ .

Figure 8. Density distribution of the difference between the consumption growth rate implied by equation (20) (recursive utility) and the one implied by equation (45) (time additive utility), when $\delta =0.17$ .

If there were no difference between the two models, the density would be degenerate at zero, with the entire mass concentrated at zero. On the contrary, the graph shows that the density is not degenerate and most of the mass is located away from zero. The results displayed in Appendix D (Tables 11 and 12) show that for two other calibrated values of the depreciation rate, $\delta =0.1$ and $\delta =0.17$ , a similar feature is obtained.