François Caron

Luke Bornn

Arnaud Doucet

Sparsity-Promoting Bayesian Dynamic Linear Models

Keywords: generalized hyperbolic, Gaussian mixture models, sparsity, dynamic regression Modèles parcimonieux, distribution hyperbolique généralisée, modèle de mélange de gaussiennes, régression linéaire dynamique

Sparsity-promoting priors have become increasingly popular over recent years due to an increased number of regression and classification applications involving a large number of predictors. In time series applications where observations are collected over time, it is often unrealistic to assume that the underlying sparsity pattern is fixed. We propose here an original class of flexible Bayesian linear models for dynamic sparsity modelling. The proposed class of models expands upon the existing Bayesian literature on sparse regression using generalized multivariate hyperbolic distributions. The properties of the models are explored through both analytic results and simulation studies. We demonstrate the model on a financial application where it is shown that it accurately represents the patterns seen in the analysis of stock and derivative data, and is able to detect major events by filtering an artificial portfolio of assets.

Introduction

Over recent years, there has been an increased number of regression and classification applications involving high-dimensional data. In these scenarios, it is common to have a large number of predictors, a number of them being irrelevant. The need to appropriately restricts the number of predictors for improved statistical efficiency and predictive abilities has generated a large body of work. In the non-Bayesian literature sparse regression analysis via penalised likelihood has become extremely popular since the seminal lasso paper [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF]; in the Bayesian literature, spike-and-slab priors have historically been favoured [START_REF] Mitchell | Bayesian variable selection in linear regression (with discussion)[END_REF][START_REF] Zhang | Lookahead and piloting strategies for variable selection[END_REF]. Unfortunately, spike and slab priors are notoriously difficult to fit, leading to a renewed interest in proposing alternative sparsity-promoting prior models.

It is well-known that the Lasso estimate for linear regression parameters can be interpreted as the MAP (Maximum A Posteriori) estimate when the regression parameters are assigned independent Laplace priors. From a Bayesian perspective, the use of MAP estimators lacks solid justification; however, although they are not sparse in the exact sense, Bayesian posterior medians are remarkably similar in value to lasso estimates [START_REF] Park | The Bayesian lasso[END_REF] and provide credible intervals which can help in guiding variable selection.

Additionally it has been observed empirically that Markov chain Monte Carlo (MCMC) mix quite well for such models [START_REF] Kyung | Penalized regression, standard errors, and Bayesian lassos[END_REF]. However, it is well-known that the lasso estimates and its Bayesian version suffer from various problems. In particular, coefficients can get shrunk towards zero even when there is overwhelming evidence in the likelihood that they are non-zero. There has been much work in the non-Bayesian and Bayesian literature to improve over this; for example, a number of sparsity-promoting nonconcave log prior distributions have been proposed which reduce bias in the estimates of large coefficients.

Recent work includes [START_REF] Griffin | Bayesian adaptive lassos with non-convex penalization[END_REF][START_REF] Caron | Sparse Bayesian nonparametric regression[END_REF][START_REF] Lee | A hierarchical Bayesian framework for constructing sparsity-inducing priors[END_REF][START_REF] Griffin | Inference with normal-gamma prior distributions in regression problems[END_REF] and Bayesian interpretations of the group lasso and elastic net estimators [START_REF] Bornn | Grouping priors and the Bayesian elastic net[END_REF][START_REF] Li | The Bayesian elastic net[END_REF][START_REF] Kyung | Penalized regression, standard errors, and Bayesian lassos[END_REF]. Although the above priors result in non-sparse posterior median and mean estimates, many arguments have been advocated in favour of their use, see e.g. [START_REF] Kyung | Penalized regression, standard errors, and Bayesian lassos[END_REF].

In the context of time series, it is of particular interest to allow for the sparsity pattern to evolve over time as a predictor which is highly relevant in a given time period may become irrelevant later on. Dynamic sparsity modelling is an important topic that has received much less attention in the literature. From a non-Bayesian perspective, several authors have proposed to adapt the elastic net, fused lasso or group lasso to accommodate dynamic models (Angelosante et al., 2009;Angelosante and Giannakis, 2009;[START_REF] Vaswani | Kalman filtered compressed sensing[END_REF][START_REF] Jacob | Group lasso with overlap and graph lasso[END_REF]. From a Bayesian perspective, dynamic spike-and-slab type models have been recently proposed: [START_REF] Nakajima | Bayesian analysis of latent threshold dynamic models[END_REF] associate to each predictor a latent process and this predictor is only included in the regression when the magnitude of its associated latent process is above a given latent threshold, whereas [START_REF] Ziniel | Tracking and smoothing of time-varying sparse signals via approximate belief propagation[END_REF] associate to each predictor a latent binary inclusion/exclusion Markov chain. We follow here an alternative approach based on the construction of dynamic sparsity-promoting priors. Similar constructions were also recently proposed independently in [START_REF] Sejdinović | Bayesian sequential compressed sensing in sparse dynamical systems[END_REF] and [START_REF] Kalli | Time-varying sparsity in dynamic regression models[END_REF]. However, the model presented here is much more flexible. It relies on a scale mixture of normal distributions where the mixing distribution is itself a generalized inverse Gaussian resulting in a multivariate generalized hyperbolic distribution. This scale mixture of normals representation can be used to derive tailored Markov chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) methods for inference.

The rest of the paper is organized as follows. In Section 2, we review the generalized hyperbolic distribution and show that it includes numerous sparsity-promoting priors used in the literature. Section 3 presents our dynamic sparsity model and establishes some of its properties. Section 4 proposes several Bayesian computational procedures to perform inference. We demonstrate the model on simulated data and an application to financial data in Sections 5 and 6.

Sparse Bayesian regression

Bayesian regression model

Consider the following standard regression model where

y = Xβ + (1)
where y is the n × 1 vector of responses, β = (β 1 , ..., β p ) T the vector of regression parameters, X is the n × p design matrix, and is the n × 1 vector of independent and identically distributed normal errors with mean 0 and variance σ 2 . In a Bayesian approach, we adopt a prior density π (β) = p j=1 π (β j ). [START_REF] Park | The Bayesian lasso[END_REF] proposed to use independent Laplace priors, motivated by the fact that lasso estimates could be interpreted as the Bayes posterior mode under this prior [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF]. Laplace priors can be expressed as scale mixture of normal distributions [START_REF] Andrews | Scale mixtures of normal distributions[END_REF][START_REF] West | On scale mixtures of normal distributions[END_REF], hence admit a hierarchical construction. Other models, based on scale mixture of normals, have been proposed in the literature [START_REF] Tipping | Sparse Bayesian learning and the relevance vector machine[END_REF][START_REF] Caron | Sparse Bayesian nonparametric regression[END_REF][START_REF] Griffin | Inference with normal-gamma prior distributions in regression problems[END_REF]. Several of these distributions can be considered as particular case of the generalized hyperbolic distribution, which we review in the next section.

Generalized hyperbolic distribution

Let β j ∈ R, and suppose the following Gaussian mixture model

β j |τ j ∼ N (µ, τ j )
(2)

τ j ∼ GiGauss(ν, δ, γ) (3)
where N (µ, σ 2 ) denotes the Gaussian distribution of mean µ and variance σ 2 and GiGauss(ν, δ, γ) is the generalized inverse Gaussian distribution (Barndorff-Nielsen and Shephard, 2001) of parameters ν, δ, γ whose probability density function is

(γ/δ) ν 2K ν (δγ) x ν-1 exp - 1 2 (δ 2 x -1 + γ 2 x) , x > 0 (4)
β j then follows a generalized hyperbolic distribution of pdf

(γ/δ) ν √ 2πγ ν-1/2 K ν (δγ) δ 2 + (β j -µ) 2 ν-1/2 K ν-1/2 γ δ 2 + (β j -µ) 2 (5)
where K a (z) is the modified Bessel function of the third kind. We write β j ∼ GH(µ, ν, δ, γ). When µ = 0, the distribution is concentrated around 0. For some values of the parameters ν, δ and γ, the pdf will be concentrated around 0 with heavy tails, which makes it a desirable prior distribution for sparse linear regression. The generalized hyperbolic distribution generalizes several distributions that have been used as sparsity-promoting priors:

(a) ν = -1/2: Normal inverse Gaussian law [START_REF] Caron | Sparse Bayesian nonparametric regression[END_REF] (b) δ = 0, ν > 0: Normal gamma law [START_REF] Caron | Sparse Bayesian nonparametric regression[END_REF][START_REF] Griffin | Inference with normal-gamma prior distributions in regression problems[END_REF] (c) δ = 0, ν = 1: Laplace law [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF][START_REF] Park | The Bayesian lasso[END_REF] (d) γ = 0, ν < 0: Student's law [START_REF] Tipping | Sparse Bayesian learning and the relevance vector machine[END_REF] All of these priors have been described as suitable prior distributions for promoting sparsity/shrinkage in Bayesian regression models. In some cases, they have been shown to give better predictive performances than spike-and-slab priors in regression [START_REF] Griffin | Inference with normal-gamma prior distributions in regression problems[END_REF]. The mixing properties of the associated The probability density functions of the generalized hyperbolic distribution for different values of the parameters are represented in Figure 1, where we see that the parameters ν, δ and γ provide significant control over the behavior of the mode and tails.

Dynamic Sparse Bayesian regression

We now consider the problem of successive linear regression models

y t = X t β t + ε t (6) where t = 1, . . . , T is a time index, ε t ∼ N (0, σ 2 I n ), X t is a n × p design matrix, I n is the n × n identity matrix and β t ∈ R p . Assume a priori independence of the different components j = 1, . . . , p π(β 1:T ) = p j=1 π(β j,1:T ) (7)
We assume that the true vector β t is sparse, and that the sparsity pattern (indices of elements with zero values) is slowly evolving over time. We first consider a simple particular case of group sparsity where the sparsity pattern is unchanged over time, and introduce the multivariate hypergeometric distribution. We then describe how to use this distribution to define models where the sparsity pattern evolves smoothly over time, while highlighting several interesting statistical properties.

Multivariate generalized hyperbolic

Assume as a simple starting case that we want the sparsity pattern to be shared across time. This can be achieved by considering a hierarchical Gaussian model where the variance is shared over time. Suppose that,

for t = 1, . . . , T β j,t |τ j ∼ N (µ, τ j Σ) (8)
where µ ∈ R p , Σ is a positive semi-definite p × p matrix, and τ j ∼ GiGauss(ν, δ, γ). Then β j,1:T ∈ R T follows the multivariate generalized hyperbolic distribution of pdf

(γ/δ) ν (2π) p/2 γ ν-p/2 K ν (δγ) q ν-p/2 K ν-p/2 (γq) (9)
where q = δ 2 + (β j,1:T -µ) T Σ -1 (β j,1:T -µ). We write β j,1:T ∼ mGH(µ, ν, δ, γ, Σ). Shared sparsity pattern over the β j,t s is obtained through the shared variance term τ j . The matrix Σ allows one to introduce correlation between variables; again, such priors have been studied in the literature on sparse models. In particular, the group lasso prior [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF][START_REF] Raman | The bayesian group-lasso for analyzing contingency tables[END_REF][START_REF] Kyung | Penalized regression, standard errors, and Bayesian lassos[END_REF] defined by

π(β j,1:T ) ∝ exp(-γ β Σ ) (10)
is a special case of the multivariate hyperbolic when ν = p+1 2 , µ is the null vector and δ = 0. Other special cases such as the multivariate normal gamma and normal inverse Gaussian have also been studied by [START_REF] Caron | Sparse Bayesian nonparametric regression[END_REF].

Statistical model

We now turn to the use of the multivariate generalized hyperbolic distribution in the modeling of data with dependent and varying sparsity structure. We are interested in defining a model for π(β j,1:T ) that introduces correlations in time both (a) in the sparsity pattern: if the vector of regressors is sparse at time t, then it is more likely to be sparse at time t + 1, and (b) in the value of non-zero coefficients. These properties will be obtained by considering a particular decomposition of the joint distribution π(β j,1:T ) with multiple overlapping groups. Consider the following decomposition of the joint distribution:

π(β j,1:T ) = T t=d π(β j,t-d:t ) T -1 t=d π(β j,t-d+1:t ) (11)
where β j,t-d:t is marginally mGH(0 d+1 , ν, δ, γ, Σ d+1 ), where d > 0, 0 d is the null vector of length d,

Σ d =          1 α . . . α d-1 α 1 . . . . . . . . . . . . . . . . . . α d-1 α d-1 . . . 1          and α ∈ [0, 1]. In particular, if ν = d+2
2 and δ = 0, then β j,t-d:t follows the group lasso distribution (10).

The model can be alternatively defined by the following d-order Markov model (d > 0)

β j,1:d ∼ mGH(0 d , ν, δ, γ, Σ d ) (12)
and for t > d

β j,t |β j,1:t-1 ∼ GH(αβ j,t-1 , ν -d/2, 1 -α 2 δ 2 + β j,t-d:t-1 2 Σ d , γ √ 1 -α 2 ) ( 13
)
where

x Σ = √ x T Σ -1
x is the Mahalanobis distance. In the case d = 0, β j,t are iid GH(0, ν, δ, γ). Using the scale mixture representation of the generalized hyperbolic distribution, the predictive distribution ( 13) can be equivalently expressed as a scale mixture of normals with latent variables τ j,t

τ j,t |β j,1:t-1 ∼ GiGauss ν -d/2, δ 2 + β j,t-d:t-1 2 Σ d , γ β j,t |τ j,t , β j,t-1 ∼ N αβ j,t-1 , (1 -α 2 )τ j,t .
The model has the following statistical properties:

RR n°7895

(a) The model is first-order stationary with

β j,t ∼ GH(0, ν, δ, γ) (b) For any h ≤ d β j,t-h:t ∼ mGH(0 h+1 , ν, δ, γ, Σ h+1 ) (c)
The parameter 0 ≤ d ≤ T -1 tunes the evolution of the sparsity pattern over time, and we have the following special cases

• d = 0, we have independence between the sparsity patterns over time and β j,t are iid GH(0, ν, δ, γ) These properties make the model very appealing for dynamic linear regression. First, by choosing the parameters ν, δ and γ based on the large literature on sparse Bayesian regression, the user can define the level of sparsity desired in the signal. For example, if δ = 0 and ν > 0 (normal-gamma case) smaller values of ν will favor sparser solutions. Second, the user will define how this sparsity pattern is going to evolve over time, from the two extreme cases d = 0 (independent sparsity pattern over time) and d = T (shared sparsity). Between those two extremes, the value of d will tune how often the time series can alternate between sparse and non-sparse periods. This effect can be seen by looking at the autocorrelation plot for β 2 t , as shown in figures 2 and 3. The shared sparsity pattern induces a minimum level of autocorrelation over the lag d, as can be seen from Figure 3 for α = 0. Finally, the parameter α tunes the correlation between non-zero coefficients, in a classical way, and we have corr(β t , β t-1 ) = α.

• d = T -1,
In Figure 4, we represent some samples from this model for different values of d and α, to show how the sparsity pattern evolves over time depending on this parameter. This figure motivates the model for use in the modeling of stock volatility. Specifically, stock prices (and trading activity) go through alternating periods of inactivity and alacrity, making the multivariate generalized hyperbolic distribution a suitable modeling choice.

The model defined by Equations ( 12) and ( 13) relies on an integer d that tunes the evolution of the sparsity pattern. One might want to consider this parameter as varying over time, and estimate this value. t from the statistical model defined by Eq. ( 12) and ( 13) with ν = 0.1, δ = 0.01, γ = 1, T = 10 6 and α = 0. Three plots are represented with d = 5, 20, 100. As α = 0, the samples (β t ) are uncorrelated, and the autocorrelation for β 2 t is due to the shared sparsity pattern. We could then put a prior on d t , e.g. a Markov model

d t |d t-1 ∼ Bin(d t-1 + 1, ρ)
where ρ ∈ [0, 1] and Bin(n, ρ) is the binomial distribution. Alternatively, any distribution with support {0, . . . , d t-1 + 1} may be used.

Algorithms

The full posterior distribution described in the previous section is intractable; we therefore present two algorithms, one which provides fully Bayesian estimates of the regression coefficients, and the other which provides approximate MAP estimates.

Approximate MAP estimation

MAP estimation requires maximization of the following objective function

) 15
It is easy to show that

β j,t |β j,t-d:t-1 ∼ GH αβ j,t-1 , ν - d 2 , δ 2 + β j,t-d:t-1 2 Σ d , γ √ 1 -α 2 (16)
and we can therefore solve (15) with an EM algorithm using the scale mixture of Gaussian representation of the generalized hyperbolic distribution [START_REF] Dempster | Maximum likelihood from incomplete data via the em algorithm[END_REF]; [START_REF] Caron | Sparse Bayesian nonparametric regression[END_REF].

We

a convex group lasso problem [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF] for which efficient algorithms exist.

Sequential Monte Carlo algorithm

While the previous algorithms conduct approximate MAP inference, we can also write a sequential Monte Carlo (SMC) algorithm to conduct fully Bayesian inference (Doucet et al., 2001). Particularly, as memory requirements prevent implementing a particle filter with 1,000,000 particles, we employ the particle independent Metropolis-Hastings algorithm [START_REF] Andrieu | Particle Markov chain Monte Carlo methods[END_REF] to approximate the full posterior π(β 1:T , d 1:T |y 1:T ).

We can use latent variables to produce efficient proposal distributions for β t for j = 1, . . . , p, τ j,t |β j,t-dt:t-1 , d t ∼ GiGauss(ν

-d t /2, 1 -α 2 δ 2 + β j,t-dt:t-1 Σ d t , γ √ 1 -α 2 ) (19)
and

β t |τ t , y t ∼ N (µ t , Σ t ) (20) with µ t = (σ 2 t D -1 τt + X t X t ) -1 (ασ 2 t D -1 τt β t-1 + X t y t ) and Σ t = (D -1 τt + X t X t /σ 2 t ) -1 , D τ = diag(τ ).
We sample from π(β t |τ t , y t ), and the weights are simply updated with

π(y t |τ t ) = N (y t ; αX t β t-1 , X T t D τt X t + σ 2 t I n ) ( 21
)
where N (x; µ, Σ) is the probability density function of the Gaussian distribution of mean µ and covariance matrix Σ evaluated at x. The sequential Monte Carlo algorithm is described in Algorithm 1, and the particle independent Metropolis-Hastings algorithm in Algorithm 2.

Algorithm 1 Sequential Monte Carlo algorithm At t = 1

• For i = 1, . . . , N • Set d (i) 1 = 0 • For j = 1, . . . , p, sample τ (i) j,1 ∼ GiGauss(ν, δ, γ) • Sample β (i) 1 ∼ N ((σ 2 1 D -1 τ1 + X 1 X 1 ) -1 X 1 y 1 , (D -1 τ1 + X 1 X 1 /σ 2 1 ) -1 ) • Compute the weights w (i) 1 = π(y 1 |τ (i)
1 ) • Replicate particles of high weights and delete particles of low weights, so that to obtain a new set of particles. For t = 2, . . .

• For i = 1, . . . , N • Sample d (i) t ∼ Bin(d (i) t-1 + 1, ρ) • For j = 1, . . . , p, sample τ (i) j,t from Eq. (19) • Sample β (i) t from Eq. (20) • Compute the weights w (i) t = π(y t |τ (i) t )
• Replicate particles of high weights and delete particles of low weights, so that to obtain a new set of particles.

Simulation Study

We now conduct a simulation study to explore the properties and performance of the statistical model. We first generate an artificial time series of "observations" (red circles, Figure 5) generated from ground truth (solid black line, Figure 5) with additive Gaussian noise. We then explore the model's performance for d = 2, 5, ν = (d + 2)/2, δ = 0, γ = .5, 1, and α = 0, .5, .9. Figure 5 shows the model's MAP estimate of ground truth for each parameter setting, as well as the true and estimated sparsity patterns. We immediately notice that the model provides shrinkage in the estimates, as controlled by γ. In addition, the ability of the model to detect sparsity is dependent on the correlation structure. For example, when the ground truth consists of alternating values +5 and -5, and α = .9, the model fits this section as being sparse, due to the lack of smoothness. We also notice that the parameter d has two major implications. Firstly, it creates smoothness and stability in the estimate of sparsity structure. Secondly, because the model is fit online and hence the model is in some sense a filter, there is a slight delay in detecting sparsity patterns, the size of which increases with d.

We now sample from the posterior distribution through the aforementioned sequential Monte Carlo algorithm, using 1000 iterations of the particle independent Metropolis-Hastings algorithm [START_REF] Andrieu | Particle Markov chain Monte Carlo methods[END_REF], each with 1000 particles. We set σ t , ν, and γ to 1, and δ to 0.01. Also, to induce moderate correlation, RR n°7895 Ground truth (black line), observations (red circles), and fitted models (blue line, ρ = 0; green line, α = .5; magenta line, α = .9). The bottom of each plot shows the estimated sparsity pattern for the ground truth and estimates, with corresponding color coding. We observe that shrinkage is controlled by λ and smoothness in the estimated sparsity pattern by d.

Algorithm 2 Particle Independent Metropolis-Hastings algorithm Initialization • Run the sequential Monte Carlo algorithm 1, sample

β (1) 1:T ∼ N i=1 w (i) T δ β (i) 1:T
and compute

Z (1) = T t=1 1 N N i=1 w (i) t At iteration m ≥ 2 • Run the sequential Monte Carlo algorithm 1, sample β * (m) 1:T ∼ N i=1 w (i) T δ β (i) 1:T and compute Z * (m) = T t=1 1 N N i=1 w (i) t • With probability 1 ∧ Z * (m) Z (m-1) set β (m) 1:T = β * (m) 1:T and Z (m) = Z * (m) , otherwise set β (m) 1:T = β (m-1) 1:T and Z (m) = Z (m-1)
we select α = 0.8, and for the temporal correlation in d, set ρ = 0.9. These choices were made to demonstrate the estimation of d, although we emphasize that depending on the circumstances and model criterion other parameter choices wide modeling flexibility, allowing practitioners to recreate several models in the literature [START_REF] Snoussi | Bayesian blind separation of generalized hyperbolic processes in noisy and underdeterminate mixtures[END_REF][START_REF] Griffin | Bayesian adaptive lassos with non-convex penalization[END_REF][START_REF] Caron | Sparse Bayesian nonparametric regression[END_REF][START_REF] Griffin | Inference with normal-gamma prior distributions in regression problems[END_REF], as well as build unique models which extend and bridge between these models. Figure 6 shows the resulting inference for 5 replicates of the model. Here we see that the estimate of d ranges between 2 and 10, dropping during time steps when the structure of the simulated time series changes. We also plot the fitted model for each replicate, where we observe that while the fully Bayesian model does not produce sparsity, it does induce shrinkage and smoothing of the process.

Modelling Stock Volatility

Stocks, as well as their derivatives, are known to alternative periods of stability and change, both on a micro and a macro scale. On a micro scale, this often occurs due to a news item or press release setting off a flurry of trading of a given asset. As an example, consider the stock price of BP oil and gas company following news of the Deepwater Horizon oil spill in 2010. Following this news, the regular day-to-day variability in the stock price increased by orders of magnitude as a constant stream of good and bad news led to an increase in trading activity. On a macro scale, this is often due to crashes, or corrections, in the market. As an example, consider the 2000 tech bubble, or the 2008 stock market crash, which both led to massive changes in the stock market as a whole.

Stock price time series are freely available from numerous sources such as Yahoo! and others, and we study a collection of stock and derivatives which we suspect would exhibit interesting effects in their prices over the period 1998 to 2011. The first stock we study is BP; as mentioned earlier, we expect to observe massive variability following the 2010 oil spill. We similarly follow OIL, iPath's S&P Crude Oil Index.

Conversely, we look at PowerShares' Crude Oil Short (SZO), to study the effect of shorting the price of crude oil. The next asset we study is XCI, the Amex Computer Technology Index, in the hopes of observing activity from the 2000 tech bubble. Next we turn to the real estate market, as measured through ProLogis (PLD), a real estate investment trust which began in 2006, with particular interest in the 2008 market crash. These five stocks, indexes, and derivatives constitute the core of our study.

We begin by plotting the monthly change in the previously mentioned stocks and derivatives in Figure 4). From this plot, it is difficult to decipher major trends from noise. In contrast, see Figure 8.

7, where we see significant volatility in the technology sector in the early 2000's, and similar volatility in all sectors during the 2008 recession and recovery. One pattern of immediate note is that of SZO, the PowerShares' Crude Oil Short, which as expected reacts contrary to the other assets during the 2008 recession.

We now attempt to model these volatilities directly, using parameters d = 0, 3, 6, 12, ν = (d + 2)/2, δ = 0, γ = 0.1, .5, 1, and α = .5. Figure 8 plots the filtered time series for the chosen ranges of γ and d, where we see that for moderate values of these parameters (namely, the two center panels), we are able to isolate significant events, such as the early 2000's tech bubble and the 2008 recession, particularly in the housing market.

Taking a closer look at the real estate market, we now consider the very practical problem of building a portfolio in the situation where one is already largely invested in the real estate market, namely through home ownership. Specifically, the casual investor who owns their own home and wishes to diversify should aim to build a portfolio with little correlation to the housing market in case of another housing crisis. Modifying the problem slightly to regress the remaining four assets against PLD, the housing index, we calculate the regression coefficients β P LD for each time series, plotting them in Figure 9. We note immediately that for small γ and d (the top left panel), it appears that all assets are highly correlated with the housing market.

As we induce shrinkage and sparsity, however, we notice that the technology index, XCI, disappears. As expected, the derivative which is short on oil (SZO) is negatively correlated with the housing market during the housing collapse in 2008. Conversely, the remaining two variables are positively correlated during the crash. Using these results, one might choose to diversify their investment in their house with technology stocks, or as an alternative to hedging the housing market directly might instead hedge the price of crude oil.

Discussion and Extensions

The dependency structure of the β j,1:T is a d-order Markov model, which is a decomposable graph structure.

The construction proposed in this paper could be generalized to any dependence structure that is given by a decomposable graph [START_REF] Lauritzen | Graphical Models[END_REF], where the joint distribution on cliques of the graph is generalized hyperbolic. This would enable one to consider dependencies, for example, on rooted trees.

In this paper, we have proposed a novel approach for conducting dynamically sparse Bayesian regression.

Built on the class of multivariate generalized hyperbolic distributions, the proposed method generalized many existing approaches for tackling this problem, while providing added modeling flexibility. Inference on this class of models may be conducted exactly using MCMC methods, in particular the particle independent Metropolis-Hastings algorithm, or approximate but sparse approximations may be built around MAP estimator, using overlapping group lasso techniques or the EM algorithm.

The proposed class of models is well-suited to modeling stock volatility data, as the structure of the multivariate generalized hyperbolic distribution induces alternating periods of large and small volatility as observed daily market fluctuations. We demonstrate how, through this class of models, one is able to isolate large-scale variation in stock price volatility to build a conservative and robust portfolio of uncorrelated assets.

Figure 1 :

 1 Figure 1: Probability density functions of the generalized hyperbolic distribution for several values of (ν, δ, γ) correspond to a Laplace, Normal inverse Gaussian and Student t distribution. The pdf of the normal distribution is also shown for comparison. (a) Behavior around 0 and (b) Tail behavior.

  the sparsity pattern is shared over time and β j,1:T ∼ mGH(0 T , ν, δ, γ, Σ T ) (d) The parameter α ∈ [0, 1] tunes the correlation between regression coefficient values at successive time steps.

Figure 2 :Figure 3 :

 23 Figure 2: Autocorrelation function for β 2t from the statistical model defined by Eq. (12) and (13) with ν = 0.1, δ = 0.01, γ = 1, T = 10 6 and (a) d = 5, (b) d = 20. For each value of d, three plots are represented with α = 0.00, 0.80, 0.95. Note that d = 0 is not represented, since the variables β t are independent in that case. (b) For d = 20, we can clearly see that the correlation for β 2 t is due to both α and the shared sparsity pattern; after lag 100, the correlation remains due to the shared sparsity pattern. For α = 0 (black line) we can clearly see a threshold on autocorrelation, which is due to the shared sparsity pattern induced by the model.

Figure 4 :

 4 Figure 4: Samples from the statistical model defined by Eq. (12) and (13) with ν = 0.1, δ = 0.01, γ = 1, T = 2000 and (a) d = 0, (b) d = 5, (c) d = 20, (d) d = T . For each value of d, three draws are represented with α = 0.00, 0.80, 0.95. (a) For d = 0, the variables β t are independent and the sparsity pattern is not shared. (b,c) For d = 5, 20, the sparsity pattern can evolve over time, and the process alternates regions close to zero and away from zero. (d) For d = T , the sparsity pattern is shared, and the β t are either all close to zero (bottom figure in (d)) or away from zero (top and middle figure in (d)).

Figure 5 :

 5 Figure5: Simulation Results. Ground truth (black line), observations (red circles), and fitted models (blue line, ρ = 0; green line, α = .5; magenta line, α = .9). The bottom of each plot shows the estimated sparsity pattern for the ground truth and estimates, with corresponding color coding. We observe that shrinkage is controlled by λ and smoothness in the estimated sparsity pattern by d.

Figure 6 :

 6 Figure 6: Results from fitting simulation data using SMC. Estimate of d over time (top). Simulated observations and fitted model for 5 replicates (bottom).

Figure 7 :

 7 Figure7: First differences of a portfolio of stocks and derivatives. Note the similarities with realizations from the model (Figure4). From this plot, it is difficult to decipher major trends from noise. In contrast, see Figure8.

Figure 8 :

 8 Figure 8: Model fit of first differences of a portfolio of stocks and derivatives. Note the effects of the early 2000's tech bubble in the technology index (XCI), the 2008 housing market crash in the real estate investment trust (PLD), and the 2010 oil spill on the BP stock.

Figure 9 :

 9 Figure 9: Regression coefficients of a portfolio of stocks and derivatives regressed on PLD. Note that as d increases, the estimation of the coefficients is stabilized.

RR n°7895

351, Cours de la LibérationBâtiment A 29

Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399


Kausik Chaudhuri

Mary M Cherical

Microeconomic Analyses of Economic

Keywords: Credit rationing, Formal Sector, India JEL Classification: O12, O20, O53

come

Introduction

Government intervention is a common phenomenon in the rural credit markets of developing countries. India was no different in this aspect. The intervention in India took the form of nationalisation of banks, bank branch expansion and the subsidisation of credit to the priority sector. The rural credit markets in developing countries are dual in nature, with the existence of the formal and informal credit markets. The informal sector, in general, charges exorbitant rates of interest on loans whilst the collateral requirement is negligible. The agricultural sector's need for cheap and timely credit necessitated government intervention, which intended to counter the predominance of the informal sector. Furthermore, the informal credit sector acted as a deterrent to the growth of the rural sector. Hence, the government emphasised on spreading banking facilities in the rural areas.

Existing research supports government intervention based on adverse selection, moral hazard, and contract enforcement problems that would affect the level of investment (Stiglitz et al., 1981). In addition, credit rationing in rural credit markets may be present due to increased market power in the hands of a few, who have access to information (Besley, 1994). However, the interventionist policy of the government has led to the misallocation of resources. This was because the interest rates were kept artificially lower than the market rates, allocation of credit went into the hands of the powerful, and political precedence took over economic ones. The issue of credit rationing in the formal credit markets arises, as the rural poor are not able to furnish collateral as a guarantee for a loan. Based on their asset holdings, households are found to be credit rationed. Bhende (1986) using data from three ICRISAT villages demonstrates this type of credit rationing based on attributes like higher education, older head of households, larger family size, and larger farm size. Evidence also exists towards credit constrained situation of rural households by the formal financial institutions (Zeller (1994), Kochar (1997), Diagne (1999), Duong and Izumida (2002), Pal (2002), Swain (2002)). Diagne (1999) discusses the determinants of household access and participation in both the formal and informal credit markets of Malawi using the concept of credit limit. The paper expounds the importance of composition of assets rather than its monetary value or land holding and reveals that formal and informal sectors are imperfect substitutes and households are credit constrained in both the sectors. The study further elucidates that households give more importance to easy access to credit rather than the actual costs involved in getting credit. This calls for the removal of the subsidised rates of interest prevailing in the markets. In this context, we note that in India, interest rates were deregulated in 1994. Before 1994, India followed the administered interest rate mechanism. 1 The importance of access to credit can arise due to various factors: firstly, in case of large seasonal fluctuations in income, it can smooth consumption requirements. Secondly, credit can be used as a source of finance for both working and investment capital for the households who do not have much savings. Finally, an emergency event such as illness can create the pressure to borrow.

In the informal sector, monitoring may fail if the distribution of returns from investments is influenced by the actions of the borrowers. Highly indebted individual may not have the incentive to work hard as a significant proportion of the return would be utilised for repayment purpose. In this case, high interest rates create a debt-overhang problem and the informal lenders may not raise the interest rate beyond a critical level. Besides, with limited funds, not all applicants may get credit. Against the backdrop of the rural credit market, this paper tries to examine the following two questions. We first deal with the joint determination of households' requirement for credit and the financial institutions decision on granting the loan. This way of modelling takes care of the sample selection bias that was inherent if we just include only those households who have applied for a loan. The second pertains to the choice of the household whether to apply for a loan and if so, then from which financial institution/s. We address these questions using data from the 54 th round of the National Sample Survey (NSS) on Consumer Expenditure and Common Property Resources conducted in January -July 1998.

The results indicate that credit rationing does exit in the formal credit market in rural-India. The households with an account in a financial institution have a higher probability of applying and receiving a loan compared to no-account households. The households who possess lower land holding are significantly credit constrained than those who possess large land holding.

Female headed household although has a lower chance of applying for a loan but the probability of receiving loan given that they have applied, are higher than their male counterpart. Livestock possession is considered as collateral that is liquid in nature. Hence, those households who own livestock have a greater chance of receiving a loan. The indices, which measure the wealth of the household, reveal that households who have a higher durable goods index have a greater chance of participating in the credit market. The presence of a financial institution in the village as well as adequate infrastructure facility increases the probability of the households demand and the supply of loans by financial institutions.

The paper is organised in the following manner. Section 2 discuses the data and the variable used in the analysis. The empirical model along with the obtained results is presented in Section 3. Section 4 presents the results and Section 5 contains the conclusion. The data for our analysis has been extracted from the Consumer Expenditure and Common Property Resources, National Sample Survey 54 th Round (January-June, 1998). 2 We have used the rural household data for the fifteen major states of India. Our sample indicates that 88.52% of the households did not apply for a loan from any of the financial institutions in the past two years before the date of survey whereas 10.35% of the households in the sample applied to at least one financial institution. 3 The number of households who applied to more than one financial institution accounted for less than 1.2% of the sample surveyed. Out of the total 7,415 households applied for loan, 20.11% of households were not granted the loan.

Our aim is to explain the credit rationing in terms of household and village level characteristics. The first set of household variable are the family size, sex of the household head, and age of the household head. The size of the family is a major determinant of borrowing by households as larger the size of the family; the greater would be their credit requirements. The average size of the family is around five in our sample. Past studies have dwelt upon the importance of women and credit and in general document the better repayment behaviour by women. Thus, the sex of the household head (dummy taking the value of one if the household head is female) is used to check for any gender specific impact. Around nine percent of the head of the households are women in the sample. To perceive the link between the age of the household head and the credit requirement the age of the household head is included. 4 In our sample, 13% of the households belong to scheduled tribe, 21% to scheduled caste and 27% to other backward caste. In absence of information on educational status, we use subscription of newspaper dummy as a proxy to capture the literacy as well as awareness of the household. Most of the sampled households are engaged in agricultural activities as their major occupation (about 56%).

The second set of household variables captures the economic condition. We include possession of livestock (dummy taking the value of one in case of possession), land holding dummies, durable good index, and household living condition index. The two indices, namely durable good index, and household living condition index are used as proxies for wealth of the household. 6 This would enable us to capture the separate effect of durable goods in addition to household living condition in obtaining a loan. The asset index of durable goods comprise of the variables that capture possession of the following gadgets -Television, Telephone, Radio, Electric Pump, and Diesel Pump. The index used to capture the household's living condition includes the following variables: source of drinking water, bathroom type, drainage arrangement for wastewater, and latrine type. 7 Livestock possession (dummy) is included as it is often used as collateral, which is liquid in nature. Hence, possession of livestock improves the chances of the household in obtaining credit. Land possession (dummies) would be indicative whether the household is in a position to furnish collateral for a loan. Following the definition of the Reserve Bank of India, we divide the land holding into three categories: small (>0-2.5 acres), medium 5 The policy of the financial institutions, in particular, the scheduled commercial banks is to lend a certain amount to the priority sector, of which, 18% is allotted to agricultural credit. The occupation type dummies would hence, indicate if the agricultural households were favoured over the other households in terms of occupational status. 6 The data on consumer expenditure cannot be used in our study due to the mismatch in the number of household surveyed in the main and special subjects' surveys. Constrained by this problem, we use the data on household assets to measure wealth of a household (Filmer et al. (2001)). In order to do so, we create the durable goods index and living condition index from a set of asset indicators using principal component analysis. 7 The mean value of the index is zero by construction. (2.51-5 acres) and large (above 5 acres) dummies. Moreover, the land holding dummies would divulge whether credit rationing exists based on the size of the land holding. In our sample about 56% of the households can be termed as small farmers.

Finally, we have used information on whether the household has an account in a financial institution or belonging to a CCS/SHG as well as on the purpose of loan. Having an account should increase the probability of a household to borrow from a financial institution where as the purpose for loan dummy would indicate whether a certain purpose of loan was given greater importance over others. About 30% of the households in our sample have an account in a financial institution.

The village level facilities are captured by the financial institutions dummies (Banks, RRB, CCS, and SHG) indicating the presence of the financial institution in the village. The infrastructure index is used to control for the level of infrastructure in the village. The educational index is used to control for the presence of educational institutions in the village. 8

Empirical Methodology

The sanction of the loan by the financial institutions is a two-stage process. Sanction can be given for only those households who have applied for a loan from a financial institution.

Given this, we estimate a model with sample selection following Heckman (1979) with respect to loan applicants to (1) banks and (2) CCS after having controlled for village level facilities. Consider the following:

i i i u X Y 1 1 1 1 + = β i i i u X Y 2 2 2 2 + = β 8
We have followed the same procedure as in case of household asset index to construct the facility indices at the village level. For construction of infrastructure facility index, we use presence of railway station, bus stop, Metalled road, post office, telegraph office and telephone booth. The presence of primary, secondary and tertiary schools is used to construct the educational index. where 'i' = (1,….., n), X ji is a 1 X K j vector of covariates, β j is a K j X 1 vector of parameters. The data is observable on Y 1i if Y 2i > 0 i.e. if the household has applied for a loan, while there is no observation if Y 2i ≤ 0 i.e. if household did not apply for a loan.

Our next model examines the choice behaviour of the household in deciding whether to apply for a loan and if they do, then from which financial institution they would seek the loan.

Hence, the household, in other words, compares the utility arising from applying for a loan from a specific financial institution/s and not applying for a loan at all. This model primarily models the choice of the household's credit requirement and is an unordered choice model, namely using a multinomial logit model (MNL). 9

Given our data, there are eight choices from which the household decides its preferenceapply for a loan from a bank, apply for a loan from CCS, apply for a loan from SHG, apply to Bank and CCS, apply to CCS and SHG, apply to SHG and Bank, apply to all and not apply for a loan. Let P 1, P 2, P 3, P 4, P 5, P 6, P 7, and P 8 be the probabilities associated with these eight choices. In this case, we take 'not apply for a loan' as the base category. Hence,

) 1 ,......., 2 , 1 ( ) exp( ) ( 1 ) ( - = = - = m j X X

Π =

In addition, the log likelihood is

ij ij m j n i P y L log log 1 1 = = ∑ ∑ =
In the multinomial logit regression model, the relative risk ratio is reported as it facilitates easy comparison across categories.

Regression Results

We report our regression results separately in terms of household variables and village characteristics first for the sample selection model and then for the multinomial model.

Sample Selection Model

Household Characteristics

The results for the sample selection model are reported in Table 1 andTable 2. The results indicate that larger the family, higher is the probability of applying for a loan from both institutions. In the main equation (after correcting for sample selection), the coefficient of family size is negative indicating that larger the family the probability of receiving a loan is lower in the case of loans from Banks. 10 Female-headed household shows less chance of applying for a loan, however, the probability of loan approval is higher for them. The coefficient of the age of the household head is positive and insignificant. The occupation type of the household has a negative and significant impact on loan approval from banks. The chances of receiving a loan from a bank improve if the household head is a woman. The probability of loan application to both agencies 10 The effect is not significant in case of loan approved by CCS. Does the economic condition matter? Our results document that it matters in case of loan application but not for loan approval. The coefficient of land possession dummy reveals interesting pattern. The coefficient is larger as the household has more lands. The index of durable goods is important, however, the same is not the case with living condition index. Similar inference is also valid regarding social group dummy. They are important in influencing the loan application equation, but not the approval equation.

Then the natural question remains: what determines the loan approval equation. Our result clearly demonstrates the role of two factors: occupational dummy (even after controlling after selection) and the purpose of the loan. Two occupational groups are definitely worse off, namely the self-employed in non-agriculture and agricultural labourer. 11 Our results also show that the probability of receiving a loan increases if the purpose of the loan is for farm business compared to any other purposes.

Village Characteristics

The coefficient of presence of cooperative credit society dummy is negative and significant in the case of loan application to banks. This indicates that CCS acts as a substitute source of credit. The same is true in case of regression equation for credit cooperative society. 12 The self-help group dummy is positive and significant indicating that the presence of the institutions in a village increases the chance of households applying for a loan from Banks. The coefficient of infrastructure index is positive and significant.

11 See the marginal effect. 12 Banks act like a substitute. The result of the multinomial logit model is presented in Table 3. As before, we report the findings in terms of household characteristics and village level variables.

Household Characteristics

The results indicate that the odds are in favour of households with an account in a financial institution. An increase in the size of the family has a favourable impact on loan application to Banks, CCS, Banks-CCS, and CCS-SHG. There is a negative and significant impact on loan application to Banks, CCS, and Banks-CCS when the household head is a woman and the older the household head the lower are the chances of applying for a loan from Banks, and SHG. The odds favour those households who have subscribed to newspaper in applying for a loan to Banks-CCS and CCS-SHG in contrast to those households who do not subscribe to newspaper.

The odds also favour households who possess livestock when compared to those who do not own livestock in applying for a loan to Banks, CCS, SHG, Banks-CCS, and SHG-Banks. The relative ratio of land holding dummies is positive and significant for loan application to Banks, CCS, Bank-CCS, and All FI's. The size of the coefficients indicates that households having larger land holding have a greater chance of applying for a loan in comparison to those who do not own land.

The odds favour those households who have a higher durable good index in applying for a loan from Banks, SHG, Banks-CCS, and All FI's. The relative risk ratio of the household index is negative and insignificant for all the choices except for SHG wherein it is significant. The result indicates that households with a higher household index are richer and are hence not part of SHG's. It also indicates that the lower income households participate in the self-help groups, as they would not have any other means of obtaining a loan. The households who derive their main income from agricultural labour have a relative risk ratio in favour of applying to Banks, SHG, CCS-SHG, and All FI's. The odds of those households whose occupation is 'other labour' have a higher probability of applying to Banks, and CCS-SHG. The odds are in favour of those households self-employed in agriculture in applying for a loan to Banks, CCS, and CCS-SHG. The social group dummies signify that the odds are against the scheduled tribe households in seeking a loan from CCS, CCS-SHG, and SHG-Banks. The odds favour the scheduled caste and OBC households in applying for a loan from banks.

Village-Level Variables

The odds of a household applying for a loan from CCS decreases with the presence of commercial banks in the village whereas, presence of CCS in the village improves the chances of the household applying for a loan from CCS in comparison to the case when these institutions were outside the village. The presence of a SHG in a village improves the probability of a household in the village to apply for a loan from Banks. The presence of SHG improves the chances of the household in applying to SHG, Banks-CCS, and CCS-SHG. The presence of CCS in the village reduces the odds of the household applying to all the financial institutions. An increase in the infrastructure index has a positive impact of loan application to Banks-CCS. The education index does not have an impact on loan application.

Conclusion

This paper examines the prevalent situation of the formal financial institutions in rural India using the data from the Consumer Expenditure and Common Property Resources, National Sample Survey 54 th Round (January-June, 1998). Our results reveal that the rural households are considerably credit constrained. The households who do not have an account in a financial these households are the family size, caste and the age of the household head. Those households who borrow for non-farm purpose exhibit a lower chance of obtaining credit compared to those households who borrow for farm business. This could be indicative of the fact that, emphasis has been given to the priority sector when the issue of lending arises. The presence of a financial institution in the village improves the probability of households seeking a loan and improves the chances of the loan being granted. The infrastructure index reveals that the villages that have a higher index would represent better infrastructure facility and hence, improve the chances of the household applying and receiving the loan. In sum, we conclude that the lack of adequacy and access to credit in the rural credit markets still exists in India.

F

  on their social groups and the same is the case with occupational dummies.5

  can use nested logit or conditional logit here. We test for the Independence of Irrelevant Alternatives (IIA) assumption using both Hausman and Small-Hsiao test. Our results indicate that IIA assumption holds for the model.

  household has an account in a financial institution. However, note except the sex, family size or the age of the head of household, the other variables are not significant in loan approval equation.

  those household self employed in non-agriculture in applying for a loan from Banks, CCS-SHG and All Financial Institutions in comparison to the 'other' households.

  lower chance of obtaining the loan. Households who are credit constrained have relatively lower land holding and they do not possess livestock. The other factors that go against

Table 1 Results for loan in case of Commercial banks

 1

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Table 2 Results for loan in case of Credit Cooperative Society (CCS)

 2

	Variables	Loan Approved by	Loan Application	Marginal
		CCS	to CCS	Effects
		Second Equation	First Equation
	Household Level variables
	Household has an account (if yes = 1)		0.732	0.119
			(0.000)***
	Family Size	0.011	0.011	0.006
		(0.335)	(0.009)***
	Sex of the head of the Household (if female = 1)	0.054	-0.234	-0.020
		(0.702)	(0.000)***
	Age of the head of the Household	0.004	0.001	0.002
		(0.073)*	(0.425)
	Household subscribes to a newspaper (if yes = 1)	-0.074	0.006	-0.028
		(0.557)	(0.889)
	Economic Condition
	Household posses livestock (if yes = 1)	0.081	0.188	0.064
		(0.318)	(0.000)***
	Land Possession Dummy
	If less than 2.5 acres	-0.105	0.341	0.019
		(0.377)	(0.000)***
	If more than 2.5 acres less than 5 acres	-0.148	0.481	0.025
		(0.319)	(0.000)***
	If more than 5 acres	0.017	0.644	0.105
		(0.918)	(0.000)***
	Index of Durable Goods	0.039	0.024	0.019
		(0.159)	(0.021)**
	Index of Living Condition	0.009	-0.015	0.001
		(0.792)	(0.259)
	Occupational Dummy
	Self employment in non-agriculture	-0.235	0.009	-0.093
		(0.105)	(0.849)
	Agricultural labour	-0.212	0.047	-0.074
		(0.110)	(0.270)
	Other labour	-0.123	0.019	-0.045
		(0.440)	(0.718)
	Self employment in agriculture	-0.161	0.246	-0.019
		(0.235)	(0.000)***
	Social Group Dummy
	Scheduled Tribe	0.019	-0.165	-0.022
		(0.898)	(0.010)**
	Scheduled Caste	-0.111	0.023	-0.039
		(0.276)	(0.536)
	Other Backward Caste	-0.028	0.035	-0.005
		(0.762)	(0.324)

Table 3 Multinomial Logit Model including village facilities

 3

		Bank	CCS	SHG	Bank-CCS	CCS-SHG	SHG-Bank All
	Variables	RRR	RRR	RRR	RRR	RRR	RRR	RRR
	Household Level
	Variables
	Household has an account	4.197	5.555	1.851	5.797	2.056	4.662	1.795
	(if yes = 1)
		(0.000)***	(0.000)***	(0.002)***	(0.000)***	(0.066)*	(0.000)*** (0.020)**
	Family Size	1.052	1.027	1.014	1.040	1.089	1.022	0.977
		(0.000)***	(0.005)***	(0.567)	(0.066)*	(0.005)***	(0.626)	(0.397)
	Sex of the head of the	0.577	0.551	0.834	0.557	0.685	1.298	1.050
	Household (if female = 1)
		(0.000)***	(0.000)***	(0.393)	(0.035)**	(0.457)	(0.535)	(0.834)
	Age of the head of the	0.996	1.000	0.989	0.997	1.005	1.002	1.006
	Household
		(0.012)**	(0.963)	(0.033)**	(0.618)	(0.671)	(0.836)	(0.275)
	Household subscribes to a	1.066	0.950	1.154	1.749	3.759	1.938	1.653
	newspaper (if yes = 1)
		(0.470)	(0.610)	(0.655)	(0.011)**	(0.028)**	(0.165)	(0.137)
	Economic Condition
	Household posses livestock	1.530	1.538	1.380	1.891	1.503	3.783	0.977
	(if yes = 1)
		(0.000)***	(0.000)***	(0.047)**	(0.001)***	(0.301)	(0.000)*** (0.933)
	Land Possession Dummy
	If less than 2.5 acres	1.270	2.134	1.365	1.281	1.034	2.035	1.942
		(0.003)***	(0.000)***	(0.171)	(0.454)	(0.943)	(0.135)	(0.025)**
	If more than 2.5 acres	1.551	2.897	1.436	2.127	0.571	1.625	2.488
	less than 5 acres
		(0.000)***	(0.000)***	(0.290)	(0.022)**	(0.441)	(0.440)	(0.004)***
	If more than 5 acres	1.745	3.783	1.796	4.078	1.301	1.353	3.066
		(0.000)***	(0.000)***	(0.106)	(0.000)***	(0.678)	(0.629)	(0.002)***
	Index of Durable Goods	1.067	1.022	1.112	1.131	1.091	1.097	1.331
		(0.001)***	(0.331)	(0.059)*	(0.006)***	(0.268)	(0.309)	(0.003)***
	Index of Living Condition	0.987	0.967	0.874	0.963	0.965	1.012	1.115
		(0.557)	(0.254)	(0.054)*	(0.574)	(0.813)	(0.935)	(0.389)
	Occupational Dummy
	Self employment in non-	2.906	1.022	1.251	1.253	19.928	1.129	1.876
	agriculture
		(0.000)***	(0.845)	(0.327)	(0.402)	(0.005)***	(0.812)	(0.013)**
	Agricultural Labour	1.757	1.151	1.573	1.157	7.539	0.574	1.998
		(0.000)***	(0.172)	(0.061)*	(0.569)	(0.063)*	(0.240)	(0.011)**
	Other Labour	1.671	1.087	1.275	0.978	6.827	1.630	1.510
		(0.000)***	(0.476)	(0.422)	(0.953)	(0.097)*	(0.447)	(0.215)
	Self employment in	2.137	1.920	1.089	1.344	13.174	0.935	0.999
	agriculture
		(0.000)***	(0.000)***	(0.728)	(0.161)	(0.015)**	(0.864)	(0.998)
	Social Group Dummy
	Scheduled Tribe	1.078	0.706	0.568	0.962	0.000	0.000	1.078
		(0.510)	(0.028)**	(0.167)	(0.911)	(0.000)***	(0.000)*** (0.808)
	Scheduled Caste	1.441	1.066	0.948	0.811	1.139	1.227	1.363
		(0.000)***	(0.477)	(0.784)	(0.324)	(0.766)	(0.597)	(0.222)
	Other Backward caste	1.125	1.114	0.756	0.878	0.635	1.131	1.004
		(0.056)*	(0.193)	(0.121)	(0.451)	(0.296)	(0.747)	(0.986)

Page 20 of 20 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  RRR implies the relative risk-ratio. Robust p-values adjusted for clustering are in parentheses. * significant at 10%; ** significant at 5%; *** significant at 1%.

	Submitted Manuscript P o F r e e r Bank CCS SHG Village Level Variables RRR Table 3 continued RRR RRR Presence of Commercial Bank inside the Village 1.174 0.744 1.187 (0.130) (0.013)** (0.525) Presence of Cooperative Bank inside the Village 0.859 1.540 1.113 (0.114) (0.000)*** (0.694) Presence of Regional Rural Banks inside the Village 0.948 0.820 1.122 (0.602) (0.134) (0.716) Presence of Self-Help Group inside the Village 1.186 1.084 2.857 (0.030)** (0.420) (0.000)*** Educational Facility Index 0.969 0.988 1.064 (0.315) (0.767) (0.394) Infrastructural Facility Index 1.035 1.052 0.994 (0.204) (0.165) (0.956) State Dummy Included Yes Observations 58599 Wald χ 2 916853.20 Prob > χ 2 0.0000 Pseudo R 2 0.7989 Log pseudo-likelihood -24500.335 Note: Page	Bank-CCS RRR 0.718 (0.220) 1.080 (0.745) 0.762 (0.337) 1.479 (0.054)* 0.940 (0.475) 1.132 (0.088)*	CCS-SHG RRR 0.549 (0.196) 0.654 (0.323) 0.779 (0.637) 2.189 (0.025)** 1.165 (0.128) 1.036 (0.811)	SHG-Bank All RRR RRR 0.914 0.862 (0.825) (0.649) 1.173 0.548 (0.622) (0.089)* 0.722 1.613 (0.418) (0.149) 1.309 1.489 (0.359) (0.227) 1.312 0.981 (0.105) (0.861) 0.924 0.937 (0.606) (0.612)
	R e v i e w
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Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Gabriella Sjögren Lindquist
email: gabriella.sjogren.lindquist@sofi.su.se

Jenny Säve-Söderbergh
email: jenny.save-soderbergh@sofi.su.setel

D81 -Criteria for Decision-Making under Risk and Uncertainty < D8 -Information and Uncertainty < D -Microeconomics, C93 -Field Experiments < C9 -Design of Experiments < C -Mathematical and Quantitative Methods, C72 -Noncooperative Games < C7 -Game Theory and Bargaining Theory < C -Mathematical and Quantitative Methods

Keywords: Individual Decision-making, Bounded Rationality, Natural Experiment, Game-Show Individual Decision-making, Bounded Rationality, Natural Experiment, Game Show JEL Codes: C93, C72, D81

This paper empirically investigates the common assumption of economic agents' capabilities to process complex mathematical problems to find optimal strategies applied in economic modeling. By exploiting a design difference in the game show Jeopardy between the U.S. and Sweden we obtain a natural experiment of individuals facing an optimization decision either having explicit information or deriving it by non-complex adding and subtracting. Given the assumption that individuals compute optimally, there should be no difference in the strategies used. Yet, the results show that even a small change in informational pre-conditions for obtaining an optimal strategy strongly alters economic-decision making.

I Introduction

An enduring controversy in economics regards the conflict between the assumption of rationality and the fact that economic agents have limited capacities to process information. Individuals are commonly assumed in theory to be able to compute their optimal strategy for a given situation, regardless of the extent of computational abilities necessary to reach the optimal decision. Such rationality of an agent is often criticized for being too strong an assumption since many maximization problems are quite difficult, which implies that people would not be able to carry them out in practice.

In this paper we explore the assumption that individuals are able to calculate an optimal strategy by taking advantage of a design difference in the game show Jeopardy between the U.S. and Sweden. The vital difference in the design is that in the U.S. game contestants are given explicit information before they act, while in the Swedish version contestants need to perform calculations to possess the same information. Given that individuals are capable of computing optimally, there should be no difference in the strategies used in the two designs. This is especially the case since acquiring the information merely requires tedious but simple addition and subtraction on the part of the player. Consequently, the difference in the design of the otherwise similar games can be used as a natural experiment. On top of that, by using a television game show, we obtain the advantage of a natural large-stake setting for decision making. 1 In addition, by exploring the design difference of the games, our paper complements earlier research on rationality by using a control and treatment approach in which only one condition is changedthe need for individual calculation. In the game show Jeopardy three participants compete in a quiz game for two successive rounds before they enter a final. In the final of the game, the contestants bid any amount of their accumulated score on their ability to answer a final question. This bid is either added to their score with a correct answer to the final question, or subtracted from it with an incorrect answer.

The winner of the game is the contestant with the highest total score. It is this final bidding decision that is the focus of this paper.

In this final bidding situation, the contestant who has the lead before the final play, i.e. the pre-final leader, is faced with two fairly straightforward strategies that he (or she) can apply to guarantee a victory. Given that the pre-final leader is assumed to prefer to win, there should be large incentives for the player to use these strategies.

The strategies are first contingent on how much the pre-final leader is ahead of the runnerup and second on the pre-final leader's knowledge of his lead. It is in this second condition that the game design differs between the two countries. In the U.S. version of the game this information, i.e. the scores, is publicly given to the contestant while in the Swedish design it needs to be derived by adding and subtracting the scores of the other contestants' correct and incorrect answers while actively participating in the game. Apart from the information aspect, and the fact that the final score is converted into each countries' currency, the games have the same strategic set-up.2 Given this similarity we may rely on the difference displayed in strategies coming from the need for Swedish players to derive the same information that is explicitly given to U.S. players. We ultimately compare the betting strategies applied between the two game designs by replicating the analysis performed by Metrick (1995) for the U.S. Jeopardy with a Swedish sample. The Swedish Jeopardy data are gathered from televised broadcasts of the show during 2002. 3 The U.S. Jeopardy data are directly based on results published by Metrick (1995). The U.S. sample thus serves as our control group with players having explicit information on the other players' scores, to be compared with our treatment group of players who have to compute the same information. Our aim, however, is not to establish any other betting strategy. We only wish to illuminate how a small change in the preconditions for finding an optimal strategy can alter maximization behavior.

In essence, we find no evidence of the same strategic behavior in our games with selfderived information as that found by Metrick with explicitly given information. Only a negligible share of adopted strategies bear similarity to those in the U.S. sample. This result is somewhat unexpected given that these strategies would have secured a victory. Given the degree of difficulty of the average question on Jeopardy and the high level of general knowledge required by the contestant even to be able to enter the game show, it is clearly the case that contestants on Jeopardy possess intellectual abilities above those of contestants on most other television game shows. 4 It is therefore surprising that such contestants are unable to keep track of the other two participants' scores in the pre-final rounds when all it involves is adding and deducting. Our results thus suggest that even when faced with a relatively simple problem most contestants show limited ability to calculate optimal strategies. Related papers are, apart from Metrick (1995), those by Bennett and Hickman (1993), Berk et al. (1996), Healy and Noussair (2004) and Tenorio and Cason (2002), who use different It is found that players do not behave rationally if the problem is hard to solve, which is interpreted in terms of bounded rationality. Our paper thus complements these findings, but also differs by using a different setting.

The paper is arranged as follows. In Section 2 the game design is explained and strategies are outlined. Section 3 presents the data. The results are given in Section 4, which is followed by a discussion in Section 5. Section 6 offers concluding remarks.

II Jeopardy Game Design and Strategies

The game show Jeopardy works as follows. After two successive rounds, in which the players accumulate scores according to their ability to answer questions 6 correctly and their ability to signal that they want to answer by pushing a button, the contestants enter the Jeopardy final.

In the final, the three players privately (i.e. hidden from the other players) bid a chosen amount of their pre-final score on a subject area in which an unknown final question will be asked. If a contestant gives the correct answer to the final question, the bid is added to the contestant's pre-final score. If the answer is incorrect, the bid is subtracted from the pre-final score. The contestant with the highest final score becomes the Jeopardy Champion and keeps a sum in the country's currency equivalent to the amount of the score (less tax). The champion is also invited back to play in a subsequent round of Jeopardy to face new contestants. 7 The 5 There are several other papers that also use TV game shows as field experiments but that test for other behaviors such as risk aversion or discrimination (Antonovics et al. 2005; Beetsma and Schotman 2001; Fullencamp et al. 2003; Gertner 1993; Post et al. 2008). 6 A special feature of Jeopardy is that the contestants are given the answer to a question and they have to give the correct question to the answer. To avoid confusion, we will use the term correct "answer" to refer to the correct question that they give. 7 If two (or all three) players tie, both (or all) keep their prize money and are invited back to the next show as Jeopardy Champions. In both countries' versions the contestants are told the score levels of the other contestants at the end of the first round. The vital difference between the U.S. and the Swedish design comes from contestants' information on the other contestants' pre-final scores after the second round. In the Swedish game, the contestants have to add and subtract the scores of the other contestants' correct and incorrect answers while they actively participate in the game. Gathering the information is thus possible by simple math, although it may be tedious. U.S. contestants are explicitly and publicly told the exact score of all the players after the second round has been played.

The strategy we employ is to compare betting strategies between the two game designs.

We therefore replicate the analysis performed by Metrick using our Swedish sample. By adopting this set-up we can analyze whether players differ in their betting strategy when the need for basic calculations in the decision process is introduced.

For the pre-final leaders in the U.S. Jeopardy, who are explicitly given information on their lead, Metrick derives two best responses depending upon the amount the pre-final leaders are ahead. 10 The first best response is found in "runaway games", in which the pre-final leaders' can secure a victory through strategic bidding. Since the final bid, 1 Y , cannot exceed the contestant's pre-final score, 1 X , then at certain relative positions a pre-final leader can guarantee a victory. This applies when the pre-final leader has a pre-final score, 1 X , twice as high as the pre-final score of the runner-up, 2 X . In these runaway games the leader can be certain of winning as long as his bid does not exceed the difference between his own pre-final score and twice the pre-final score of the runner-up, i.e.

2 1 1 2X X Y - < , [1]
here defined as runaway bids.

The second best response is found in "shut-out games", where the pre-final leader can make a bid ensuring a victory if he (or she) gives the correct answer to the final question. In these games, the pre-final leader's score is not high enough to secure a victory but Metrick shows that the pre-final leader has the possibility to make a bid that shuts-out the runner-up with a correct answer. This is the smallest possible bid that ensures a sole victory if he answers correctly, i.e.

1 2

1 2 1 + - = X X Y , [2]
here defined as a shut-out bid. An example of such a game is when the pre-final leader has 10,000 and the runner-up has 7,000 (and the third player only has 1,000 and can be ignored), whereupon the maximum amount that the runner-up can win is 14,000. The smallest possible bid the prefinal leader can make to secure a non-tied victory for a correct answer is 4,001. 11 Note that it is the smallest possible bid that ensures a victory with a correct answer and that bids above this level also guarantee victory but at the cost of losing more if the answer is incorrect.

III Data

In Table I some summary statistics of the Swedish and U.S. games are shown. The Swedish sample comprises 206 shows, but in 11 of the shows the second runner-up had a negative (or 0)

11 In Metrick's analysis it is shown that the focal bid is to ensure a sole victory, not a tie. Hence we analogously define a shut-out bid as a bid that secures a sole victory. There are no noteworthy differences in general between the two countries. Swedish prefinal leaders are only slightly more likely to state the correct answer to the final question compared with U.S. players. In Sweden a few more games start with the pre-final leader having a large lead.

IV Results

Metrick reports that, in the U.S. version of Jeopardy, none of the 110 pre-final leaders in runaway games made a bid larger than the runaway bid threshold. Of these, 24 made a bid exactly equal to, or U.S.$1 less than, the runaway bid threshold. That is, all U.S. pre-final leaders who had the opportunity secured their victory by basing their strategies on the precise score level of the runner-up.

In our Swedish sample, 64 of the 196 pre-final leaders had the opportunity to make runaway bids. The frequencies of runaway bids are summarized in Table II. None of the Swedish pre-final leaders made a bid at, or, like U.S. players, SEK 1 less than the runaway bid threshold and only 17 made runaway bids.

The distribution of bid deviations from the runaway bid threshold for pre-final leaders in 9 range and among these the absolute deviations are high. Note that in the positive range the bid ensures no victory, either with a correct answer or with an incorrect answer. The average deviation from the runaway bid threshold for the Swedish players is SEK 2,584, which is far above 0. In a one-sided t test, we can also reject that the average deviation is 0, or negative, at the 1 percent level.12 Hence, in our sample, where players had to perform calculations to receive information on which they could form their strategy, only 27 percent secured a victory, while 100 percent did so in Metrick's sample, where the players have explicit, instead of derived, information.

In the games where the pre-final leader's score is not high enough to secure a victory, i.e.

in the shut-out games, Metrick finds that the modal bid for pre-final leaders is the shut-out bid and that this is made in over half of the games. Metrick defines this as a focal bid. In our sample of shut-out games, none of the 132 pre-final leaders chose an exact shut-out bid, and 2 percent placed a bid within a negative deviation of SEK 1, accepting a tie. The statistics are given in Table II. In addition, Metrick finds that 85 percent of the shut-out bids in U.S. Jeopardy are played between a perfect shut-out bid and a positive deviation of 1,000. In contrast, 17 percent of the Swedish players made a bid within this interval. 10 Swedish players is SEK 3,218, which is clearly far from 0. Moreover, as a result of a one-sided t test, we can reject that the average deviation is equal to 0 at the 1 percent level.

V Discussion

In runaway games, the pre-final leader can win with certainty given that he uses the information on the runner-up's score. However, only 26 percent of participants who had the possibility in our sample behaved rationally and made runaway bids compared with 100 percent in the U.S.

Jeopardy. In shut-out games, only 2 percent used the modal strategy adopted by 57 percent of the U.S. players.

In this section we discuss whether these deviations from optimal simply reflect that Swedish players are inexperienced. In his paper, Metrick also analyzes the behavior of the runners-up. He finds that the U.S. runners-up bid non-optimally given the optimal play of the U.S. pre-final leaders and that experience is one explanation for the deviance of non-optimal behavior. Moreover, when analyzing The Price is Right, Bennett and Hickman (1993), Berk et al. (1996), Healy and Noussair (2004) and Tenorio and Cason (2002) find that more experienced individuals behave more "rationally", or in line with what would be optimal, than inexperienced players.

To capture such an experience effect we estimate, similar to Metrick, a logit regression on the probability of placing a runaway bid on the number of games attended. For the Swedish prefinal leaders we find no difference between experienced and inexperienced players, as displayed in Table III. A similar regression but on the probability of placing a bid equal to the shut-out bid or larger, defined as a shut-out interval bid, obtains the same conclusion. Note that all these bids secure a victory for a correct answer but have a higher cost to the player in terms of a larger loss 11 if the answer is incorrect. We interpret this absence of an experience effect as the players finding the problem too complex to solve even when they become more experienced.

Further, we explore whether runaway bids or shut-out interval bids are used more often by pre-final leaders when the strategies are easier to compute. Metrick finds that runners-up in games where the optimal strategy is more difficult to calculate are less likely to use optimal strategies. The difficulty of calculating a strategy is measured by the score spread between the pre-final leader and the runner-up. It is assumed that in games with a large score spread it is less tedious for the pre-final leader to keep track of the runner-up's score since the pre-final leader has answered most of the questions himself. Thus, in games with a higher score spread, we would see that Swedish pre-final leaders behave more like those in the U.S. In the runaway games there is an additional positive effect on the probability of using runaway bids with a large score spread as the cut-off level of a runaway bid declines with a higher score spread. For the shut-out games, on the other hand, there is a positive effect with a smaller score spread since, if the pre-final leader has a sense of being very close to the runner-up, he knows that to be able to win the game he has to place a higher bid to win with a correct answer.

Similar to Metrick, we construct a variable, RATIO, which is the ratio between 2 X and 1 X , to capture the score spread between the pre-final leader and the runner-up. As suggested by the estimates displayed in Table III, we find, in line with our hypothesis, a positive significant correlation between the probability of placing a runaway bid and the score spread. 13 For the shutout interval bids we find the negative effect on the score spread to be dominant. Consequently, we do not argue that players necessarily are totally unable to keep track of each others' scores - 13 If we contemplate that it is easy to compute the score differences, then strategic play ought to be more likely in the games with only two players. In these games the second runner-up performed so badly that the score can be ignored.

In our 11 two-player games the probability of using runaway bids is 0. For the shut-out games the probability of making a shut-out interval bid is 0.11. Hence even when faced with an easier calculation, players still do not play according to runaway or shut-out strategies.

VI Concluding Remarks

By analyzing the strategies adopted in the two otherwise similar games, we obtained a test of whether the need for self-deriving information matters for the decision structure. As the strategies of those based on explicit information cannot be replicated by players who need to self-derive the same information, we can establish that altering the pre-conditions of economic decision making can have a large impact. Therefore, assuming that individuals are equipped with a perfect capacity to calculate optimal strategies in any given situation is not unquestionable in general.

Considering the fairly non-complex mathematical skills, mainly addition, required for the Jeopardy game, the assumption of individuals having an almost perfect capacity to calculate seems even more difficult to justify for more complex economic decisions. Note: Standard errors are given in parentheses. The Runaway Dummy takes the value 1 for pre-final leaders who played runaway bids and 0 for all pre-final leaders who had the possibility to play a runaway bid. The Shut-out Interval Dummy takes the value 1 for players who played an exact shut-out bid or higher and 0 for all pre-final leaders who had the possibility to play a shut-out bid. Note that none of the players played an exact shut-out bid; only 2 players were close with bids yielding a tie. a EXPERIENCE is the number of consecutive shows the contestant has participated in. b RATIO is the ratio between X2 and X1.

F

1 1 2 1 2 1 - + - < X X Y 16 0.12 8 0.03 1 1 2 1 2 1 - + - = X X Y (accepting a tie) 2 0.02 26 0.09 Bid = Shut-out bid 1 2 1 2 1 + - = X X Y 0 0.00 135 0.48 Bid > Shut-out bid 100 1 2 1 2 1 2 1 1 2 + + - ≤ < + - X X Y X X 1 0.01 40 0.14 000 , 1 1 2 100 1 2 1 2 1 1 2 + + - ≤ < + + - X X Y X X 19 0.14 40 0.14 000 , 1 1 2 1 2 1 + + - > X X

  of the game show The Price is Right to evaluate the use of optimal strategies. 5

  games a winner can play is five successive rounds.8 The first and second runners-up receive non-monetary prizes. 9

  score and thus could not play in the final round. Since the 2 remaining players find themselves in a different strategic situation from that of players in a 3-player game, we exclude these 11 shows from our main sample. Altogether, we have 585 observations. The U.S. sample comprises 393 games and 1150 observations.

  runaway games is shown in Figure I. At 0, the pre-final leader made a bid exactly at the runaway bid threshold. In the positive range the bid is higher than the runaway bid threshold and in the negative range all the bids are runaway bids. The majority of bids, 73 percent, are in the positive

Figure

  Figure II shows the distribution of deviations from a shut-out bid. As in the runaway bids'

  have a sense of being far behind or very close -just that they do not base their strategies on the exact score level of the other players.

FiguresFigure I Figure

 I Figures

  : The Swedish sample was collected from video-recorded transmissions of Jeopardy during 2002. The U.S. sample is based on published results inMetrick (1995).

	Tables
	Table I Summary Statistics
	Swedish sample	U.S. sample
	Pre-Final Leader Correct	63%	54%
	Percentage of Runaway Games	33%	28%
	Percentage of Shut-out Games	67%	72%
	Total Observations	585	1150
	Total Games	196	393
	o r F o r F o r
	P P P
	e e e
	e r e r e r
	R R R
	e e e
		v i e v i e v i e
		w w w
	14

Note

Table II

 II Behavior of Pre-final Leaders in Runaway Games and Shut-out Games

	RUNAWAY GAMES:	1 X ≥	2X	2	Swedish sample	U.S. sample
													Obs.	Freq.	Obs.	Freq.
	Mean bid deviation from Runaway bids	2,584***
													(4,644.3)
	Bid ≤ Runaway bid threshold
	1 Y	<	X	1	-	2 2 X	-	1				17	0.27	86	0.78
	1 Y	=	X	1	-	2 2 X	-	1				0	0.00	20	0.18
	1 Y	=	X	1	-	2X	2						0	0.00	4	0.04
	Bid > Runaway bid threshold
	1 Y	>	X	1	-	2X	2						47	0.73	0	0.00
	Total											64	1.00	110	1.00
	Allowing for a SEK 500 error
	1 Y	≤	X	1	-	2 2 X	+	500			22	0.27
	1 Y	>	X	1	-	2 2 X	+	500			59	0.73
	Total											81	1.00
	Allowing for a SEK 1,000 error
	1 Y	≤	X	1	-	2 2 X	+	1000			24	0.24
	1 Y	>	X	1	-	2 2 X	+	1000			76	0.76
	Total											100	1.00
	SHUT-OUT GAMES:	1 X ≤	2X	2
	Mean bid deviation from Shut-out bids	3,218***
													(3,270.7)
	Bid < Shut-out bid-1

Table III Logit

 III Y Note that in the Swedish sample there are 196 pre-final leaders in 195 games. In 1 game there is a tie between the first player and the runner-up and hence there are 2 pre-final leaders in this game. Standard deviations are given in parenthesis. Regression on the Probabilities of Bidding Runaway Bids or Shut-out Bids or Higher for the Swedish Sample of Pre-final Leaders

		Runaway	Shut-out Interval
		Dummy	Dummy
		(1 = YES,	(1 = YES,
		0 = NO)	0 = NO)
	EXPERIENCE a	-0.19	0.05
		(0.32)	(0.26)
	RATIO b	-10.95	5.63
		(3.46) ***	(1.99)***
	CONSTANT	2.76	-6.32
	LR Chi2 Pr ( > Chi) F o r No of Obs.	(1.23) 13.03 0.002 64	(1.71)*** 12.03 0.010 132
	P
	e
	e r
		R
		e
		v i e
			w
		94	0.71	34	0.12
	Total	132	1.00	283	1.00
		16

Note:

The maximum amount a contestant can gain is SEK (Swedish kronor)

283,200 (U.S.$1 ≈ SEK 8) in 1 show and SEK 1,416,000 in 5 subsequent shows. This requires certain strong restrictions on the evolution of the game, however. The highest gain attained from a single show is SEK 88,200 and SEK 179,900 for 5 subsequent shows. The average gain per contestant is SEK 7,151 and the average gain per show is SEK 13,906 for all the Swedish broadcasts (Jeopardy Historia 2003).

Note that U.S.$1 is approximately equal to SEK 7.5, implying that the stakes are higher in the U.S. However, empirical evidence from game-show data shows that the value of the stakes does not affect the probability of players using optimal behavior. For example,[START_REF] Tenorio | To Spin or Not to Spin? Natural and Laboratory Experiments from The Price is Right[END_REF] find within the game show The Price is Right that the size of the stakes does not affect the individuals' probability of behaving rationally. Similarly,[START_REF] Healy | Bidding Behavior in the "Price is Right" Game: An Experimental Study[END_REF] find no difference when comparing outcomes from The Price is Right with prizes ranging from U.S.$1,000 up to U.S.$60,000, with a laboratory setting of the same game having prizes ranging from U.S.$2 and U.S.$100. However, some laboratory experiments using other settings have shown that stakes may matter, see for example[START_REF] Kachelmeier | Examining Risk Preferences under High Monetary Incentives: Experimental Evidence from the People's Republic of China[END_REF].

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

The transmissions are Jeopardy shows from the spring of 2002, the summer of 2002 (video-recorded reruns from the fall of 2001) and the fall of 2002.

Prior to participating in the televised game, all the participants perform a test.Page

of 18 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

The maximum number can in fact exceed five, if the winner is selected for the Jeopardy Champion contest that takes place every season. The selected contestants are the three players with the highest winning score attained that season. In the data we have one contestant who appears six times.

The values of the non-monetary prizes are approximately the same for both runners-up.

As pointed out by Metrick, the game-theoretical equilibria of the Jeopardy final are very complex. Like Metrick, we make no attempt to test any game-theoretical predictions. We refer the reader to[START_REF] Metrick | A Natural Experiment in "Jeopardy![END_REF], where a stylized version of the U.S. Jeopardy can be found.

If we allow the pre-final leader to make an error of SEK 500 in calculating the runner-up's score due to the need to derive the information on the other players' scores, the percentage who made runaway bids does not increase. Allowing for an error interval of SEK 1,000 we obtain a percentage of 24. Note that the percentage falls, since the number of individuals who made runaway bids increases by 2, but the number of individuals with the possibility of making runaway bids increases by 19. Furthermore, analyzing the deviations from runaway bids shown in FigureI, we observe that the distribution is skewed to the right. If we were to believe that players were simply making calculation errors, then the distribution of deviations should be left of 0 with the errors symmetrically distributed around 0.