Habit, Long-Run Risks, Prospect? A Statistical Inquiry

The goal of this article is to fill a void in the literature. There are, to our knowledge, no head-to-head, statistical (i.e., likelihood based or asymptotically equivalent) comparisons of asset pricing models from macro/finance. This paper fills the void. The asset pricing models considered are the habit persistence model of Campbell and Cochrane (1999), CC hereafter, the long-run risks model of Bansal and Yaron (2004), BY hereafter, and the prospect theory model of Barberis, Huang, and Santos (2001), BHS hereafter. There are two reasons for this choice: These three models are arguably the leading contenders and the authors describe their computational methods precisely enough to permit replication of their results.

The need for a statistical comparison of asset pricing models is underscored by the ongoing debate between advocates of the long-run risks and habit models. Beeler and Campbell (2008) claim that the long-run risks model is rejected by historical data on the basis of the predictability of excess returns, consumption growth, dividend growth, and their respective volatilities by the price to dividend ratio. Bansal, Kiku, and Yaron (2009) argue that the long-run risks model provides adequate predictability results when using a vector autoregression (VAR) based on consumption growth, price to dividend ratio, and the real risk-free rate. Bansal, Kiku and Yaron also argue that the habit model provides counterfactual predictability results for the price to dividend ratio when using lagged consumption growth as a regressor. Because the argument is based on the selective use of statistics by the advocates, it can be continued indefinitely without resolution. In contrast, the likelihood of model output contains all the information in these models. Therefore, likelihood-based inference, in principle, resolves the debate definitively.

We know of only one other study that attempts a head-to-head statistical comparison of asset pricing models: Bansal, Gallant, and Tauchen (2007), BGT hereafter. BGT compared the habit model to the long-run risks model using frequentist methods. Their methods could not distinguish between the two models because frequentist nonnested model comparison methods require abundant data. Abundant data are not available in macro/finance. The typical sampling frequency used to calibrate and assess macro/finance models is annual and there are only about 80 annual observations available on the U.S. economy. The papers cited above use annual data. BHS insist that annual is the only frequency that is appropriate to their model. Using more abundant higher frequency data to compare models is not an option. They were not designed to explain high-frequency data.

Failing to achieve a definitive statistical result, BGT proceeded to compare the models using the more traditional methods of macro/finance, which consist of enumerating some moments, evaluating them both from the data and from a model simulation, and comparing, often without taking sampling variability into account. On the basis of such comparisons, BGT conclude that the long-run risks model is preferred. Of these comparisons, they relied mostly on the fact that the habit model provides counterfactual predictability results for the price to dividend ratio when using lagged consumption growth as a regressor.

In addition to the fact that the BGT comparison, in the end, was not statistical, there are other concerns. BGT did not actually compare the models proposed by CC and BY. They modified them to impose cointegration on macrovariables that ought not diverge. They also used a general purpose method to solve them; specifically, a Bubnov–Galerkin method (Miranda and Fackler, 2002, 152–3). Our view is that fairness dictates that one should use the model that was actually proposed by the originator in comparisons, not a modified model, and that one should use the same solution method that was proposed. To state our view succinctly, the model is the simulation algorithm proposed by the originator; it is not the mathematical equations that suggested the algorithm.

The data we use are annual, per capita, real, U.S. stock returns, consumption growth, and the price to dividend ratio from 1925–2008. The comparisons are for the periods 1930–2008 and 1950–2008. The data from 1925–1929 are only used to prime recursions because they are of lower quality than the data from 1930 onward. The data are plotted in Figure 1. Note in the figure that consumption growth is far more volatile in the 1930–1949 period than in the period 1950–2008. It turns out that the difference in consumption growth volatility dramatically influences results.

Gallant and McCulloch (2009), GM hereafter, introduced a Bayesian method for fitting a model from a scientific discipline (scientific model for short) for which a likelihood is not readily available to sparse data. They synthesize a likelihood by means of an “auxiliary model” and simulation from the “scientific model.” GM used the term “statistical model” for auxiliary model. We do not because the consonance between scientific model and statistical model causes confusion and also because we call into question the premise on which GM chose the term statistical model.

In the GM framework, the auxiliary model must nest the scientific model for the methodology to be logically correct. GM argue that it is better to use an auxiliary model that represents the data well, even if it does not nest the scientific model. We conduct a sensitivity analysis employing six auxiliary models of differing complexities to determine if the choice of auxiliary model makes a difference to our results. The six models are shown in Table 1. Model f₁ is closest to that used by GM; f₅ is the nesting model for univariate (stock returns alone) and bivariate (consumption growth and stock returns) data.

We use the protocol set forth in GM to establish nesting, which, briefly, is as follows. The models shown in Table 1 are the beginning of a sequence (whose progression is described in Subsection 2.4) that, if continued, is dense for the space in which the scientific model must lie. Therefore, what needs to be done to find a nesting model is to select the correct truncation point. Here we use the Schwarz (1978) criterion to select it followed by some diagnostic checks that GM discuss.

Increasing the dimension of a multivariate time series often simplifies the conditioning structure. That happens here: f₀ is the nesting model for trivariate (stock returns, consumption growth, and price to dividend ratio) data. We check our f₀ results using models f₁ through f₃. Computations to fit a nonlinear model to data that does not identify it well are often unstable, as they are for models f₄ and f₅ when fitted to trivariate simulations. Therefore, we do not consider f₄ and f₅ for trivariate data. The prospect theory model puts its mass on a two-dimensional subspace. This violates the GM regularity conditions. Therefore, we do not consider the prospect theory model for trivariate data.

We find that the computational methods that GM proposed are not sufficiently accurate to compare the habit, long-run risks, and prospect theory models. This is due to the fact that the auxiliary model f₅ that nests these three is more complex than the auxiliary models that GM considered. A contribution of this paper is a refinement of GM's methods that increases accuracy to the point that auxiliary models as complex as f₅ can be used in applications.

1 MODELS CONSIDERED

The intuitive notions behind any consumption-based asset pricing model are that agents receive wage, interest, and dividend income from which they purchase consumption. Agents seek to reallocate their consumption over time by trading shares of stock that pay a random dividend and bonds that pay interest with certainty. This reallocation is done for the purpose of insuring against spells of unemployment, providing for retirement, and so on. Trading activity enters the model via the constraint that an agent's purchase of consumption, bonds, and stock cannot exceed wage, interest, and dividend income in any period. When applying the model to a national economy, consumption and dividends can be used as the driving processes instead of wages and dividends because wages can be recovered by subtracting dividends from consumption. (Someone must own the stock so the dividends must be received, while for bonds someone pays interest and another receives so there are no net interest receipts.) Agents are endowed with a utility function that depends on the entire consumption process. The first-order conditions of their utility maximization problem determine a map from present and past values of the driving processes to the present price of a stock and a bond. These models are simulated by first simulating the driving processes and then evaluating the map that determines stock and bond prices. For each model, we shall describe the driving processes and the utility function, leaving a description of the algorithm for computing the map to the cited literature.

Throughout this section, lower case denotes the logarithm of an upper case quantity, for example, c_t = log(C_t), where C_t is consumption during time period t, and d_t = log(D_t), where D_t is dividends paid during period t. The exceptions are the geometric risk-free interest rate r_ft = − logP_{f,t − 1} and the geometric stock return inclusive of dividends r_dt = log(P_dt + D_t) − logP_{d,t − 1}, where P_{f,t − 1} is the price of a discount bond at the beginning of time period t that pays $1 with certainty at the end of period t,P_{d,t − 1} is the price of a stock at the beginning of time period t, and P_dt its price at the end.

1.1 The Habit Persistence Model

The prior is Equation (1). For the habit model, the scale factor used for φ and δ is 0.001 rather than the 0.1 shown in Equation (1) to overcome an identification problem. The MCMC chain will not mix when the scale factor for φ and δ in Equation (1) is 0.1 because a move in φ can be nearly exactly offset by a move in δ. The value 0.001 is the largest value for which MCMC chains will mix. Because of the first term of the right-hand side of Equation (1), (1) is not an independence prior; in simulations from the prior no correlation between parameters is zero.

Measures of location and scale for the prior and posterior distributions of habit model parameters are shown in the top panel of Table 2. The prior and posterior densities of the risk-free rate, equity premium, equity returns, and the standard deviation of equity returns are plotted in the left column of Figure 2. Overall, Table 2 and Figure 2 suggest that the prior is sufficiently informative to fill in where data are sparse but it allows the data to move the posterior where data are informative.

Where differences in the three models are the most obvious visually is in their out-of-sample forecasts for the next five years. The mean posterior forecast for the habit model, computed as described in Subsection 2.6, is plotted in the left column of Figure 3. The habit model predicts an end to the current recession in 2009 and return to steady-state growth by 2010. This is dictated by Equations (2), (7), and (8), which imply that annual consumption growth for the habit model is a first-order autoregression with an autogression parameter of about 0.25 (Working 1960). Stock returns are predicted to be high in 2009 with a return to steady-state returns by 2013.

1.2 The Long-run Risks Model

It is so richly parametrized that identification would have to come from the prior even when data are abundant because half of the models in Table 1 have fewer parameters than θ. The time increment is one month. Aggregation of monthly {c_t,r_dt,r_ft}_{t = 1}^12N to the annual frequency {c_t^a,r_dt^a,r_ft^a}_{t = 1}^N is by means of Equations (7) through (10).

The prior is Equation (1). The autoregressive parameters ρ and η cause problems. The solution method proposed by BY degrades as ρ and η increase from their published values. Because the degradation is continuous in ρ and η, there is no logical threshold that one can impose on ρ and η to completely prevent degradation. Our solution to this problem is to set the scale factor for ρ and ν in Equation (1) to 0.01 rather than 0.1 and attenuate the tails for ρ and ν above 0.995.

Measures of location and scale for the prior and posterior distributions of the parameters of the long-run risks model are shown in the middle panel of Table 2. The prior and posterior densities of the risk-free rate, equity premium, equity returns, and the standard deviation of equity returns are plotted in the middle column of Figure 2. As for the habit model, Table 2 and Figure 2 suggest that the prior is sufficiently informative to fill in where data are sparse, but it allows the data to move to the posterior where data are informative.

The mean posterior forecast for the long-run risks model is plotted in the middle column of Figure 3. The long-run risks model predicts an end to the current recession in 2010 and a slow increase in the growth rate thereafter. Stock returns are predicted to be at their steady-state values over the entire forecast period. A flat response of asset returns to a consumption growth and asset return shock is a structural characteristic of the long-run risks model. It is due to the fact that stochastic volatility is autonomous and therefore not affected by consumption growth and asset return shocks. The stochastic volatility process is the factor that affects the risk premium in the long-run risks model.

1.3 The Prospect Theory Model

The driving processes for the prospect theory model are as follows:

Identification requires the prior for most of the auxiliary models considered. The time increment is one year: Simulate directly and set r_dt^a = r_dt, and r_ft^a = r_ft.

Measures of location and scale for the prior and posterior distributions of the parameters of the prospect theory model are shown in the bottom panel of Table 2. The prior and posterior densities of the risk-free rate, equity premium, equity returns, and the standard deviation of equity returns are plotted in the right column of Figure 2. As previously, Table 2 and Figure 2 suggest that the prior is sufficiently informative to fill in where data are sparse, but it allows the data to move the posterior where data are informative.

The mean posterior forecast for the prospect theory model is plotted in the right column of Figure 3. The prospect theory model predicts steady-state growth throughout the forecast period. This is dictated by the fact that annual consumption growth for the prospect theory model is independent and identically distributed. Stock returns are predicted to be double their steady-state value in 2009, reach steady-state by 2011, and remain at steady-state thereafter.

2 INFERENCE FOR GENERAL SCIENTIFIC MODELS

We describe the Bayesian methods proposed by GM and the modifications that we found necessary. Public domain code implementing the method for the auxiliary models in Table 1 and a User's Guide are available at http://econ.duke.edu/ webfiles/arg/gsm.

2.1 Estimation of Scientific Model Parameters

The difficulty is computing the implied map g accurately enough that the accept/reject decision in an MCMC chain (step 5 in the algorithm below) is correct when f is a nonlinear model. We describe the algorithms that we have found effective next.

We use N = 5000, which requires 60,000 monthly simulations in the case of the habit and long-run risks models. Results (posterior mean, posterior standard deviation, etc.) are not sensitive to N; doubling N makes no difference other than doubling computational time. By accident we once set N = 60,000 in the prospect theory model; this also made no difference. It is essential that the same seed be used to start these simulations so that the same θ always produces the same simulation.

GM run a Markov chain {η_t}_{t = 1}^K of length K to compute that solves Equation (24). There are two other Markov chains discussed below so, to help distinguish among them, this chain is called the η-subchain. While the η-subchain must be run to provide the scaling for the model assessment method that GM propose, the that corresponds to the maximum of over the η-subchain is not a sufficiently accurate evaluation of g(θ) for our auxiliary models. This is mainly because our auxiliary models use a multivariate specification of generalized autoregressive conditional heteroskedasticity (GARCH) that Engle and Kroner (1995) call BEKK. Likelihoods incorporating BEKK are notoriously difficult to optimize. We use as a starting value and maximize Equation (24) using the BFGS algorithm (Fletcher 1987, 26–40). This also is not a sufficiently accurate evaluation of g(θ). A second refinement is necessary. The second refinement is embedded within the MCMC chain {θ_t}_{t − 1}^R of length R that is used to compute the posterior distribution of θ. It is called the θ-chain. (We use R = 25,000 past the point transients have dissipated.) Its computation proceeds as follows.

The θ-chain is generated using the Metropolis algorithm. The Metropolis algorithm is an iterative scheme that generates a Markov chain whose stationary distribution is the posterior of θ. To implement it, we require a likelihood, a prior, and transition density in θ called the proposal density. The likelihood is Equation (23).

The prior may require quantities computed from the simulation used to compute Equation (23). Our prior requires r_f^a. The sequence is available from the simulation. The risk-free rate for the prior is the average . (For the habit and prospect models, r_ft^a is constant over the simulation.) Quantities computed in this fashion can be interpreted as the evaluation of a functional of the scientific model of the form Ψ:p(·|·,θ)↦ψ. Thus, our prior is a function of the form π(θ,ψ). However, the functional ψ is a composite function, θ↦p(·|·,θ)↦ψ, so that π(θ,ψ) is ultimately a function of θ only. Therefore, we will only write π(θ,ψ) when it is necessary to call attention to the subsidiary computation p(·|·,θ)↦ψ.

Let q denote the proposal density. For a given θ, q(θ,θ^*) defines a distribution of potential new values θ^*. We use a move-one-at-a-time, random-walk proposal density that puts its mass on discrete, separated points, proportional to a normal. Two aspects of the proposal scheme are worth noting. The first is that the wider the separation between the points in the support of q the less accurately g(θ) needs to be computed for α at step 5 of the algorithm below to be correct. As an example, the long-run risks model is not sensitive to the risk aversion parameter γ so that values of γ could be separated as much as one-fourth without making any difference to the usefulness of the θ-chain with respect to inference regarding economics. A practical constraint is that the separation cannot be much more than a standard deviation of the proposal density or the chain will not move. Our separations are typically one-eighth of a standard deviation of the proposal density. In turn, the standard deviations of the proposal density are typically no more than the standard deviations in Table 2 and no less than one order of magnitude smaller. The second aspect worth noting is that the prior is putting mass on these discrete points in proportion to π(θ). Because we never need to normalize π(θ), this does not matter. Similarly for the joint distribution f(y|x,g(θ))π(θ) considered as a function of θ. However, f(y|x,η) must be properly normalized as a function of y, at least to the extent that Equation (24) is computed correctly.

The algorithm for the θ-chain is as follows. Given a current θ^o and the corresponding η^o = g(θ^o), we obtain the next pair (θ^′,η^′) as follows:

1. Draw θ^* according to q(θ^o,θ^*).

2. Draw according to p(y_t|x_{t − 1},θ^*).

3. Compute η^* = g(θ^*) and the functional ψ^* from the simulation .

4. Compute .

5. With probability α, set (θ^′,η^′) = (θ^*,η^*), otherwise set (θ^′,η^′) = (θ^o,η^o).

We use K = 200. All that is important is that transients have died out by the time the midpoint K/2 of the η-subchain has been reached and that η_K/2 and η_K are nearly uncorrelated.

We compute posterior probabilities using a method that requires one to save the values θ^′,ℒ(θ^′),π(θ^′,ψ^′) available at step 5. It also requires that these same values for a chain that draws from the prior for θ be saved. To draw from the prior, replace α at Step 4 by .

The algorithm for the η-subchain is as follows. We use a move-one-at-a-time, random-walk proposal density with continuous support. Given the current η^o, obtain the next value η^′ in the chain as follows:

1. Draw η^* according to q(η^o,η^*).

2. Compute .

3. With probability α, set η^′ = η^*, otherwise set η^′ = η^o.

Draws from the prior are also required. This is done by putting

2.2 Relative Model Comparison

Relative model comparison is standard Bayesian inference although there are a few details that need to be discussed to connect it to Subsection 2.1.

One computes the marginal density, ∫∏_{t = 1}ⁿf(y_t|x_{t − 1},g(θ))π(θ)dθ, for the three scientific models p₁(y|x,θ₁), p₂(y|x,θ₂), p₃(y|x,θ₃) with respective priors π₁(θ₁), π₂(θ₂), π₃(θ₃) using method f₅ of Gamerman and Lopes (2006, section 7.2.1). The advantage of that method is that knowledge of the normalizing constants of f(·|·,η) and π(θ) are not required, and it appears to be accurate in tests that we conducted. The computation is straightforward because the relevant information from the θ-chains for the prior and posterior are available after completion of the computations discussed in Subsection 2.1. It is important, however, that the auxiliary model be the same for all three models when the computations in Subsection 2.1 are carried out. Otherwise the normalizing constant of f would be required. One divides the marginal density for each model by the sum for the three models to get the probabilities for relative model assessment.

Note that what one is actually doing is comparing the three models f(y|x,g₁(θ₁)), f(y|x,g₂(θ₂)), f(y|x,g₃(θ₃)), with respective priors π₁(θ₁), π₂(θ₂), π₃(θ₃). This is an important observation. Inference is actually being conducted with likelihoods ∏_{t = 1}ⁿf(y_t|x_{t − 1},g₁(θ₁)), ∏_{t = 1}ⁿf(y_t|x_{t − 1},g₂(θ₂)), ∏_{t = 1}ⁿf(y_t|x_{t − 1},g₃(θ₃)), not ∏_{t = 1}ⁿp₁(y_t|x_{t − 1},θ₁), ∏_{t = 1}ⁿp₂(y_t|x_{t − 1},θ₂), ∏_{t = 1}ⁿp₃(y_t|x_{t − 1},θ₃). If f nests all p_i, that is, if Equation (22) holds, then the former and the latter are the same. If not, the matter needs consideration. In GM's application, they give two examples. In the first, the presence or absence of GARCH in the auxiliary model makes a dramatic difference to habit model parameter estimates. In the second, changing the thickness of the tails of the auxiliary model makes no difference. They argue on the basis of common sense and their examples that what is actually required is that the auxiliary model fit the observed data, not that it nests p. That is why they use the term statistical model for f. However, their argument is not a proof. We examine this issue in Section 4.

The realization that what one is actually doing is comparing the three models f(y|x,g₁(θ₁)), f(y|x,g₂(θ₂)), f(y|x,g₃(θ₃)), with respective priors π₁(θ₁), π₂(θ₂), π₃(θ₃), allows us to perform a change of measure to η-space and illustrate relative model comparison graphically. In η-space, the prior π_i for model i restricts η to a manifold ℳ_i = {η∈ℋ:η = g_i(θ_i),θ_i∈Θ_i} with each η = g_i(θ_i) in ℳ_i receiving prior weight π_i(η) = π_i(θ_i) (recall that Θ_i is discrete). Think of this manifold as a line in η-space. This is shown graphically in Figure 4 for two hypothetical models. Although conceptually a line, what is plotted in Figure 4 has an area with gray fill to represent the density of the prior along the line. A density-weighted integral along the line would have approximately the same value as an integral over the area shown. Also shown in Figure 4 are the likelihood contours of a hypothetical auxiliary model f(y|x,η). The marginal density for model i is ∫_ℳi∏_{t = 1}ⁿf(y_t|x_{t − 1},η)π_i(η)dη, which is approximately the integral over the gray areas for the hypothetical models in Figure 4. As shown in the figure, the likelihood is larger over ℳ₂ than over ℳ₁, which implies the same for the integrals over ℳ₂ and ℳ₁. Thus, the second model has the higher marginal likelihood and is therefore preferred.

2.3 Absolute Model Assessment

The scientific model can be viewed as a sharp prior on f that restricts the posterior distribution of η to lie on the manifold ℳ. Think of it as a line in η-space. If this prior is relaxed, the line becomes a region with volume in η-space. Relaxation can be indexed by a scale parameter κ. As κ increases, the size of the region increases and posterior for η will move along a path toward the likelihood of the data under f. Figure 5 is an illustration. One can select waypoints κ_i along this path, view them as the discrete values of a parameter, assign them equal prior probability, and compute their posterior probability. If waypoints near ℳ receive high posterior probability, then the data support the scientific model. If waypoints far from ℳ receive high posterior probability, then the data do not support the scientific model. The idea is illustrated graphically in Figure 5 for the case when a model is rejected and in Figure 6 for the case when a model is accepted. The formal development proceeds as follows.

We add the additional assumption that the auxiliary model is identified and has more parameters than the scientific model. This assumption implies that g ^{− 1}(η) exists on ℳ. If the scientific model is identified g ^{− 1}(η) will map to a single point; if not, g ^{− 1}(η) will be a set.

Choose three (for specificity) values κ₁, κ₂, and κ₃, ordered from small to large. Consider f under priors π_κ1, π_κ2, and π_κ3 to be three different models and compute the posterior probability for the three models with each having prior probability of 1/3. That is, the pair (f(·|·,η),π_κ(η)) is considered to be a model and the posterior probability of each κ choice is proportional to the marginal likelihood ∫∏_{t = 1}ⁿf(y_t|x_{t − 1},η)π_κ(η)dη.

If the posterior probability of model κ₁ is small, that is evidence against the scientific model. Conversely, if it is large, that is evidence in favor of the scientific model.

2.4 The Auxiliary Model

The observed data are y_t for t = 1,…,n. We first discuss the case where y_t is a multivariate. Lagged values of y_t are denoted as x_{t − 1}. For auxiliary models f₀ through f₄,x_{t − 1} = y_{t − 1}. For auxiliary model f₅,x_{t − 1} = (y_{t − 1},y_{t − 2}).

In our specification, R₀ is an upper triangular matrix, P and V are diagonal matrices, and Q is scalar; max(0,x) is applied elementwise. This is the BEKK form of multivariate GARCH described in Engle and Kroner (1995) with an added leverage term (33). In computations, max(0,x) in Equation (33) is replaced by a twice differentiable cubic spline approximation that plots slightly above max(0,x) over (0,0.1) and coincides elsewhere. Auxiliary model f₀ has term (30) only, f₁ has terms (30), (31), and (32), and f₂ through f₅ have all four terms.

The density h(z) of the iid z_t is the square of a Hermite polynomial times a normal density, the idea being that the class of such h is dense in Hellenger norm and can therefore approximate a density to within arbitrary accuracy in Kullback–Leibler distance (Gallant and Nychka 1987). The density h(z) is the normal when the degree of the Hermite polynomial is zero, which is the case for auxiliary models f₀ through f₂. For model f₃, the degree is 4. For models f₄ and f₅, the degree is 4, but the constant term of the Hermite polynomial is a linear function of y_{t − 1}. This has the effect of adding a nonlinear term to the location function (29) and the variance function (30). It also causes the higher moments of h(z) to depend on y_{t − 1} as well.

The univariate auxiliary models are the same as the above but μ_{xt − 1} in Equation (29) has dimension one and becomes the location function of a first order autoregression and Σ_{xt − 1} in (30) has dimension one and becomes a GARCH(1,1) with a leverage term added.

2.5 Diagnostic Checks

The idea behind diagnostic checking is straightforward: If one has compared two scientific models (p₁,π₁) and (p₂,π₂) using the same auxiliary model f(·|η) and the fit of (p₂,π₂) is preferred, then one can examine the posterior means (or modes) and of f(·|η) corresponding to the two fits to see which elements changed. The same is true for absolute model assessment. If one fits (f,π_κ1) and (f,π_κ2) and concludes that (f,π_κ1) fails to fit the data, then one can examine the changes in the elements of the posterior means (or modes) of f(·|η) corresponding to the two fits to see which elements changed.

Table 4 is an example.

There is a caveat. The t_i are often informative but are subject to the same risk as the interpretation of t-statistics in a regression, namely, a failure to fit one characteristic of the data can show up not in the parameters that describe that characteristic but elsewhere due to correlation (colinearity). Nonetheless, despite this risk, inspection of the t_i is often the most informative diagnostic available.

The methods proposed here are likelihood methods which means that at the conclusion of an estimation exercise a transition density that represents the data under the fitted model is available. The most useful are f(y|x,g(θ)) with θ set to the posterior mode from a fit of (p,π) and f(y|x,η) with η set to the posterior mode from a fit of (f,π_κ). One can apply standard diagnostics to these transition densities such as comparative plots as in Figure 7.

2.6 Forecasts

A forecast can be viewed as a functional Y:f(·|·,η)↦υ of the auxiliary model that can be computed from f(·|·,η) either analytically or by simulation. If f(·|·,η) nests the scientific model p(·|·,θ) then, due to the map η = g(θ), this forecast can also be viewed both as a forecast from the scientific model and as function of θ. As such, it can be computed at each draw in the θ-chain for the posterior and the posterior mean, mode, and standard deviation obtained. Similarly for draws from the prior. An example is Figure 3.

3 HABIT, LONG-RUN RISKS, PROSPECT?

Table 3 presents the posterior probabilities for a relative model comparison and an absolute model assessment of the three asset pricing models fitted to the three data series over the two sample periods.

For the univariate series, all three models fit reasonably well and none of them strongly dominates. Our interest in the univariate case is due to the fact that this conclusion depends on the choice of auxiliary model. We explore this issue in Section 4.

Our interest in the trivariate series stems from fact that BGT, Beeler and Campbell (2008), and Bansal, Kiku, and Yaron (2009) argue that the main difference between the habit model and the long-run risks model occurs with respect to predictability regressions that involve the price to dividend ratio series.

It is difficult to properly adjust dividend payouts for stock repurchases and other distortions caused by tax policy so as to make the measured data resemble the theoretical construct of asset pricing models. It is apparent from Figure 1 that the price to dividend series does not look like the realization of a stationary process. An informal check on the ability of a model to explain dividends is to simulate it for ten thousand years and see if the best match (with respect to a variance weighted mean squared error metric) of any contiguous segment of the simulation to the data looks like Figure 1. It does not. For all models and for both sample periods, the best match cannot produce the hump seen in the third panel of Figure 1.

As mentioned earlier, because the prospect theory model puts its (conditional) mass on a two-dimensional subspace, it cannot be fit to the trivariate series. However, the informal check just described is possible and the prospect theory model fails more definitively under it than either the habit or long-run risks model.

With respect to the remaining two models, in the relative model comparison presented in Table 3, the long-run risks model is dominant over the 1930–2008 period, whereas the habit model is dominant over the 1950–2008 period. Both models fail to fit the data in the absolute model assessment. Interestingly, the failure of the habit persistence model in the 1950–2008 period is not as stark as for the long-run risks model. These results are for the nesting model f₀.

The fundamental objective of asset pricing models is to explain the relationship between asset prices and consumption. From this perspective, dividends can be treated as an internal construct that need not have any corresponding observable counterpart. Therefore, the bivariate results presented in Table 3 are the most important substantively.

As seen in Table 3, in the relative model comparison, the long-run risks model is dominant over the 1930–2008 period, whereas the habit model is dominant over the 1950–2008 period. In the absolute model assessment, the habit model fails in the 1930–2008 period, and the prospect theory model fails in the 1950–2008. These results are for the nesting model f₅.

It is of interest to determine why the habit persistence model fails to fit the bivariate series using the diagnostic checks described in Subsection 2.5. For this purpose, it is more informative to use the simpler auxiliary model f₁ rather than the nesting model f₅. Because the habit model has seven parameters and f₁ has twelve, this is a legitimate choice for the habit model.

Table 4 presents the diagnostics for the habit model. In the table the fit of (f₁,π_κ) with κ = 0.1 is compared to the fit with κ = 10 for the bivariate series over the period 1930–2008 and over the period 1950–2008. It is clear from the table what the problem is. The largest diagnostic, t = − 4.98, indicates that P₁₁, which is the feedback of consumption growth into its own volatility, is too small in absolute value. The habit model fails to put enough conditional heteroskedasticity into the consumption growth process. The problem disappears in the 1950–2008 period because, as seen by comparing the entries for κ = 10 across the row for P₁₁, the conditional volatility in the data drops substantially.

Figure 7 plots the conditional volatility of the three models over the 1930–2008 period. The solid line in Figure 7 is the long-run risks model, which is the most correct of the three according to the relative model comparison in Table 3. The shaded region shows ±1.96 posterior standard deviations about this line. All three models track the conditional volatility of stock returns about the same after taking the standard deviations into account. Where they differ is in how they track the conditional volatility of consumption growth and the conditional correlation between consumption growth and stock returns. The same plot for the 1950–2008 period (not shown) looks qualitatively similar: agreement in the middle panel and disagreement in the other two.

Plots of the conditional mean of consumption growth (not shown) indicate that the habit model and long-run risks model agree over the 1930–2008 period except over the years 1930–1940. They both track the data moderately well. Because, as was seen in Figure 3, the conditional mean of consumption growth for the prospect theory model must plot as a straight line, it does not track well. All models disagree in plots (not shown) of the conditional mean of stock returns, with long-run risks plotting as a straight line; all track the data poorly.

4 SENSITIVITY ANALYSIS

There is much experience with the data shown in Figure 1. That experience suggests that about the richest model one would be willing to fit to these data is a model with one-lag VAR location, GARCH scale, and normal innovations. The exact specification one gets using standard model selection procedures, such as upward F-testing, is sensitive to the sample period used. One can get slightly richer or coarser specifications. It is fair to say that the consensus view is that a one-lag VAR location, GARCH scale, and normal innovations is the richest model one ought to entertain, which is model f₁ of Table 1.

We do not discuss the trivariate series in this section other than to remark that for it, f₀ is the nesting model, results for f₀ are reported in Table 3, conclusions do not change under auxiliary models f₁ through f₃, and computations for models f₄ through f₅ cannot be undertaken because simulations do not identify them (BFGS becomes unstable). Similarly, we do not discuss results for the bivariate consumption growth and stock returns data because the results shown in Table 3 are the same for all models in Table 1. Finally, we do not discuss absolute model assessment because there is no logical requirement that the auxiliary model nest the scientific model for the purpose of absolute model assessment. All that is logically required is that the auxiliary model have more parameters than the scientific model. Other than that, one is free to choose an auxiliary model as judgment suggests.

For the univariate (and bivariate) series, a model that will nest the three scientific models that we consider has the following characteristics: a two-lag linear conditional mean function with a one-lag nonlinear conditional mean term added to it, a one-lag GARCH conditional variance function with a one-lag leverage term and a one-lag nonlinear conditional variance term added, and a flexible innovation distribution that permits fat tails and bumps. We denote this model by f₅. It is the last of the six in Table 1. GM found the same to be true for the habit model, except that they used data from 1933–2001 with the years 1930–1932 used to prime recursions. They dismissed f₅ out of hand as absurd and worked with f₀ and f₁. They did verify that a fat-tailed innovation distribution did not change results. Using model f₀ most closely corresponds to calibration as customarily implemented in the macro/finance literature. The sufficient statistics for f₀ are the mean and variance of y_t and the first-order autocorrelations. One is, effectively, finding parameter values that best match three moments for univariate data, nine for bivariate, and eighteen for trivariate.

As discussed in Subsection 2.1 and in GM, the logically correct view toward using f₁, which fits the data, instead of f₅, which nests the scientific model, is that it is not the likelihood of the scientific model that is being used. It is some other likelihood. Therefore, it is not the scientific models that are actually being estimated and compared. Another point of view is the argument advanced by GM that using a sensible auxiliary model is akin to method of moments estimation. One only asks that the scientific models match certain features of the data and allows them to ignore others. What to do? About all one can do is try a battery of auxiliary model specifications and see what happens.

Table 5 displays the results for the relative comparisons for the univariate stock returns data over the periods 1930–2008 and 1950–2008. There is considerable sensitivity to specification of the auxiliary model over the 1930–2008 period. Conclusions are affected by the choice of auxiliary model. Our view is that, because there can be sensitivity to auxiliary model choice, and because one is not actually comparing scientific models if the auxiliary model is not nesting, it is best to use the nesting auxiliary model in general, which is f₅ in this instance.

5 CONCLUSION

We used Bayesian statistical methods proposed by GM to compare the habit persistence asset pricing model of CC, the long-run risks model of BY, and the prospect theory model of BHS. This comparison fills a void in the literature.

We undertook two types of comparisons, relative and absolute, over two sample periods, 1930–2008 and 1950–2008, using three series, trivariate (consumption growth, stock returns, and the price to dividend ratio), bivariate (consumption growth and stock returns), and univariate (stock returns). The prior for each model is that the ergodic mean of the real interest rate be 0.896 within ±1 with probability 0.95 together with a preference for model parameters that are near their published values.

For the univariate series and for both sample periods, the models perform about the same in the relative comparison and fit the data reasonably well in the absolute assessment.

For the bivariate series, in the relative comparison, the long-run risks model dominates over the 1930–2008 period, while the habit persistence model dominates over the 1950–2008 period; in the absolute assessment, the habit model fails in the 1930–2008 period, and the prospect theory model fails in the 1950–2008 period.

For the trivariate series, in the relative comparison the long-run risks model dominates over the 1930–2008 period, while the habit persistence model dominates over the 1950–2008 period; in the absolute assessment, both the habit model and the long-run risks model fail in both periods. The prospect theory model cannot be fitted to a trivariate series because it puts its (conditional) mass on a two-dimensional subspace.

The estimator proposed by GM is a simulation-based estimator. Simulations from a scientific model, which here is either the habit model, the long-run risks model, or the prospect theory model, are used to determine a map η = g(θ) from the parameters θ of the scientific model to the parameters η of an auxiliary model f(y_t|x_{t − 1},η), where y_t is the observed data and x_{t − 1} are lags. Thereafter, ℒ(θ) = ∏_{t = 1}ⁿf(y_t|x_{t − 1},g(θ)) is used whenever a likelihood is required. Theory requires that the auxiliary model nest the scientific model. GM argue that one is better served by an auxiliary model that represents the data well. We undertook a sensitivity analysis and recomputed our results for six auxiliary models ordered by complexity. The first produces estimates that mimic values obtained by methods customarily employed in macro/finance. The second represents the data. The sixth nests the three scientific models considered. We find that results can be sensitive to the choice of auxiliary models. Most importantly, results can differ between the model that represents the data well and the model that nests the scientific model. In view of this difference and the fact that theory supports the latter, our view is that the nesting auxiliary model ought to be used. Our substantive conclusions are based on the nesting model.

We found that the computational methods that GM proposed are not sufficiently accurate to compare the habit, long-run risks, and prospect theory models. A contribution of this paper is a refinement of GM's methods that allows comparision of these three models.

References

Author notes