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Economic variable selection

Regression plays a key role in many research areas and its variable selection is a classic and major problem. This study emphasizes cost of predictors to be purchased for future use, when we select a subset of them. Its economic aspect is naturally formalized by the decision-theoretic approach. In addition, two Bayesian approaches are proposed to address uncertainty about model parameters and models: the restricted and extended approaches, which lead us to rethink about model averaging. From objective, rule-based, or robust Bayes point of view, the former is preferred. Proposed method is applied to three popular datasets for illustration.


Introduction
Model selection with subsequent prediction is a classic and major problem in statistics. In the context of regression analysis, model selection is often equated with variable selection, to be accomplished in one of many ways, including classical hypothesis test of full and reduced models (e.g., Vuong (1989)), use of an information criterion such as AIC, BIC, or DIC (Akaike (1998) for a reprint of the original paper published in 1973 for AIC, Schwarz (1978) for BIC, and Spiegelhalter et al. (2002) for DIC), evaluation through some form of crossvalidation (e.g., Gelfand et al. (1992) and Gelfand and Dey (1994)), or Bayesian versions of tests based on the Bayes factor (Kass and Raftery (1995)). Subsequent prediction follows from either a separate re-fit of the data as with model selection followed by a least squares fit, or is integrated into a cohesive framework involving selection and prediction as with many of the recently-developed penalized likelihood methods.
Bayesian methods provide a distinct approach to model selection and prediction, as they are based on a cohesive modelling framework that allows one to simultaneously describe and work with their uncertainty across models and over parameters within a model. Its main applications in model selection are the hierarchical approach (Mitchell and Beauchamp (1988) and George and McCulloch (1993)) or the stochastic search approach (Hans et al. (2007) and Fouskakis and Draper (2008)). These methods follow the usual route from prior distribution through data to posterior distribution, with inference to follow. Model selection follows from inference designed to minimize incorrect model selection while prediction follows inference to minimize forecasting loss. This approach separates modelling from inference, facilitating for example, Barbieri and Berger (2004) to distinguish model selection from variable selection. It has also led to an explosion of literature on model averaging, such as Min and Zellner (1993), Madigan et al. (1995), Raftery et al. (1997), Draper (1995), Brown et al. (2002), and Yu et al. (2011), whose benefits are now well-documented.
The split between modelling and inference has generated a novel approach to parsimony within Bayesian circles which may be characterized as "fit in a large model space, make inference in a small model space (Walker and Gutiérrez-Peña (1999), MacEachern (2001), and Hahn and Carvalho (2015)). This approaches moves parsimony from modelling to inference. It seeks to construct a model that reflects the full complexity of the problem and, if little benefit is shown for some variables (aspects of the model), to move to a simpler form as part of inference. In this work, we explicitly bring an economic question into the mix-namely the cost of predictors-and pursue a path suggested by the decision theoretic formulation of the model/variable selection and prediction problem in regression. This version places our focus on two main questions: 1. Prediction, accounting for the cost of predictors. In a typical setting, predictors have costs associated with them. They cost money, take time to collect, take effort to model, or consume computational effort. These costs are real, and obtaining a slightly better prediction rule at a much higher cost or much more slowly may not be worthwhile.
2. Model uncertainty. The goal of model selection is often taken to be consistent model selection, or identification of the set of predictors with nonzero coefficients in the regression model. The economic formulation of the prediction problem suggests that a slightly inferior (in the traditional sense) model may provide a better model for practical use. This suggests a re-examination of the role of consistency in model selection.
See also Clyde and George (2004) for recent approaches to model uncertainty.
Consideration of the cost of predictors and formulation of model selection as a decision problem has appeared in the literature. Lindley (1968) argues forcefully for Bayesian methods and for a full decision-theoretic formulation of the problem. Further authors have mentioned this issue (see, for example, Brown et al. (1999) and Hahn and Carvalho (2015)). Fouskakis et al. (2009a) and Fouskakis et al. (2009b) also take the cost into account in different directions.
In this work, we seek to reconcile economic considerations with current practice in Bayesian model selection and model averaging. We find that this perspective provides strong commentary on current practice, we present several reasons to believe that current practice is generally reasonable, and we identify settings where improvements can be made.
In all, we find that Bayesian model averaging is a valuable technique but that care should be taken to its implementation. This paper is organized as follows. Two methodology is laid out in the next section.
Section 3 discusses which is more reasonable from several aspects. After that, we formalize the methodology with data in Section 4. Three data sets are used to illustrate our method in Section 5. Section 6 points to future directions and concludes the paper.

Methodology
Predictive regression modelling is often formulated as a decision problem, and it can be argued that this formulation underlies Bayesian model averaging (BMA). The traditional formulation of the problem is driven by a predictive loss of the form L(y, y(x)) = (y− y(x)) 2 , where y is a response to be predicted and y(x) is the predicted value associated with predictors x. Using standard models and integrating over the conditional distribution of the future y, this loss becomes a loss taking parameter and action as arguments, namely The variance term in equation (1) does not depend on the decision rule and hence can be ignored in determination of the optimal rule.
Our focus is on Bayesian procedures, and the Bayes rule is the Bayesian's optimal decision rule. It is typically constructed from the Bayesian posterior conditional viewpoint (Berger (1985)), by moving from prior distribution to posterior distribution and then choosing the action to minimize posterior expected (against the posterior distribution) loss.
BMA focuses on the setting where the prior distribution cuts across slices of the parameter space that are naturally described as models. In the case of linear regression with a set of p potential predictors, across the entirety of R p for the predictors' regression coefficients. A model is defined by the set of non-zero regression coefficients, and the set of 2 p potential models partition R p . The prior distribution on these coefficients is of mixed form. It typically assigns positive probability to each element of the partition-that is, to each subset of R p that corresponds to a model. The support of the prior is the entirety of each element, leading to an overall support of all of p. The prior distributions that underlie BMA are thus seen to be of slightly non-standard form, but they are prior distributions.
From this perspective, BMA follows directly as a standard Bayesian procedure. Pass from prior distribution to posterior distribution via Bayes Theorem. Once arriving at the posterior distribution, find the optimal (posterior) action. In this case, the action happens to be expressed as a summary of the model-averaged posterior distribution, or, for squarederror loss, as model-averaged posterior predictive means. BMA is nothing more (nor less) than sound application of Bayes Theorem and choice of an appropriate action. As such, it inherits all of the optimality properties of Bayesian inference.
Following Lindley (1968), we modify the original predictive loss to include the cost of data acquisition, modelling and processing, including the cost to purchase information (as in credit history for a customer), a cost of time (as in the delay in obtaining results from a medical lab test), cost in processing time (as in variables that are computationally expensive in conjunction with their use in a model), or other.
The next subsections consider a complete formulation of the problem and provide the "no-data solution" which describes the posterior Bayes action. We then describe a class of decision rules that allow the analyst to select which variables to pay for and to proceed with inference.

Decision with cost
The basic decision involves two sets of possibly overlapping and possibly null sets of predictors. The analyst must decide which to purchase, knowing the state of nature θ. For this decision, it is important to consider their costs as well as their predictive adequacies.
Let L(θ, f ; x) be the adequacy measure for x for a single future case. A simple measure would be the expected squared predictive loss, which is defined as where φ is the corresponding parameter vector when x is used as the predictor, p(y | x, φ) is the conditional probability density function derived from the state of nature, and f (x) is the action, specified for every value of x. For squared-error loss, the loss-minimizing action Let c(x) be the cost for x for a single future case. The cost depends on the set of predictors, but it does not depend on the values of those predictors. Without knowledge of predictors, a typical choice is a function of the number of predictors. See Section 5 for specific forms of the cost function.
In keeping with this formulation, we integrate the adequacy over a distribution for x, The distribution of covariates is not affected by the decision about which covariates to purchase. This is reasonable when regression is on analyzing some scientific facts.
The total cost, or the negative utility, from purchasing predictors x is expressed as the sum of L * (θ, f ; x) and c(x). It is better to purchase In general, the best predictor to purchase is chosen by solving the minimization problem: for all possible sets of predictors, x. If all predictors are free of charge, it is clear that the best combination of predictors is the one that minimizes the loss.
In this work, motivated by large samples in the future, we focus on the asymptotic performance of the method. The asymptotic evaluation is based on the limiting average total cost for a set of predictors and model. The typical conditions for consistency of parameter estimates for Bayesian regression models ensure that the model-specific fit will These conditions include full support of the prior distribution and tail conditions on the distribution of the predictors to ensure limited leverage of individual cases.
Remark 1. We note that our formulation relies on continued updating of our prior as future data accrue. Without this continued updating, our predictions would be further limited by the accuracy of an initial data set. The model-specific prediction rules would then not tend to the "best" prediction rules, even as sample size grew.

No data problem-two approaches
In this subsection, the state of nature is unknown. Its uncertainty is specified as the form of distribution about model parameters and about models. Let x be the k purchased predictors, and let w denote the p − k unpurchased predictors. The predictors may or may not be relevant to predict the response, and we expect future data of the form (x, Y ) to reveal the relationship between the response and the purchased predictors. There are two main approaches to provide forecasts for future Y as a function of the future covariate x.
The restricted approach. The restricted approach confines us to the small world of predictors x and response Y . BMA applied to this world results in model averaging across 2 k potential models, with individual predictors in x either active or not.
The extended approach. The extended approach considers the large world of models determined by predictors x and w for the response Y . BMA applied to this world results in model averaging across 2 p potential models, with individual predictors in x and w either active or not.
The first approach makes use of information only on purchased predictors. The second approach makes use of information on both purchased and unpurchased predictors. Information on the unpurchased predictors is available through the conditional distribution of the unpurchased predictors given the purchased (and maybe less expensive) predictors.
It can be considered as an extreme of imputation in the missing value problem, where all cases are missing for some predictors (see also Boone et al. (2011)). The measurement error model also has the similar structure, in that the true value is unobserved.
The predictive loss for both approaches is where h(·) is the action as a function of potential predictors.
The restricted approach removes w from the problem, restricting h to be a function of x alone, and it averages over a reduced set of models. This allows us to marginalize over w, leading to the following expression of the loss, where x γ is a subset of purchased predictors, Γ is a set of all possible γs, and π(M γ ) is the probability on the event M γ that x γ is relevant to the response.
The extended approach marginalizes the loss over w by its conditional distribution, where w λ is a subset of unpurchased predictors, Λ is a set of all possible λs, and M γλ is the event that x γ and w λ are relevant to the response. In either approach, the optimal Bayesian action minimizes the sum of predictive loss and cost of predictors.
Both of these approaches can be implemented with standard computational methods.
The restricted approach is standard BMA based on the purchased predictors. The extended approach is easiest to follow if we assume to know the joint distribution of potential predictors. In this case, the unpurchased predictors are merely missing data, to be imputed In the no-data context, the model probabilities are given. Standard specifications of prior distributions yield the prior predictive means when all predictors in a model are purchased. When one or more predictors in the model have not been purchased, the prior predictive mean may be expressed as Model uncertainty is taken into account by averaging above mean over all possible models with respect to the prior model probability. When working with a set of training data where both x and w are observed, we update the prior distribution with the full likelihood to obtain the posterior distribution. The Bayes action is the no-data action derived from the posterior distribution.

Choice of approach
One central question is whether the restricted approach or the extended approach is to be preferred. Our first take on this question is motivated by the subjective Bayesian viewpoint expressed, for example, in Savage (1972). He constructs Bayesian methods from the principles of rational behavior. This leads him to the notion of personal probability, and along with it, the ability to specify a prior distribution on unknown parameters (tied to Y | φ). The same argument allows one to specify a prior on on models and a prior on the distribution of w | x. This provides a complete description of uncertainty over models, parameters within a model, and missing predictors. Coupling this with standard results from decision theory (e.g., Berger (1985)) which state that the Bayes risk is the minimum possible risk when the parameter follows a given distribution and that the Bayes rule achieves the Bayes risk, we arrive at the usual Bayesian destination. Update the prior distribution with the likelihood for the data to obtain the posterior distribution, and use this posterior distribution for inference. In other words, the extended approach is preferred from the subjective view.
The implications of this choice run contrary to mainstream Bayesian practice. Consider a standard BMA problem where one has a set of predictors, say x and a response Y . The usual practice is to apply BMA to the set of all 2 k models. While this may appear to agree with the preceding paragraph, we can certainly envision further unobserved predictors w that may well be connected to the response at a low cost. The extended approach averages over these predictors as well, with the analyst's prior beliefs governing the relationship between x and w and the extended set of model probabilities.
This leads us to ask why BMA is practiced in its current form. Objective Bayesian methods along with their close cousin, rule-based Bayesian methods, provide a counterpoint to the subjective Bayesian perspective. The typical BMA implementation is far from subjective. Rather than using elicitation procedures to carefully specify a prior distribution across models and, for each model, a prior distribution over the parameters within the model, one resorts to a rule to determine the prior distribution. The rule may assign a set probability to each model of a given size, and it may routinely specify the distribution on the parameters given the model. Popular rules include the conjugate priors on model parameters along with the uniform prior model probability (Raftery et al. (1997)), the benchmark prior (Fernández et al. (2001)), and the mixture of g priors (Ley and Steel (2012)). See Steel (2017)  Rule-based Bayesian methods are motivated in much the same fashion as objective Bayesian methods, but more scope is allowed for choice on the part of the analyst. They maintain many of the benefits of objective Bayesian methods, allowing for easy communication of the prior distribution and a compact description of the reasoning behind the prior distribution (specification of the rule). They are often amenable to a global sensitivity analysis where the rule used to generate the prior distribution is perturbed. Yet, they are not restricted to the principles that underlie objective Bayesian analyses. These rule-based methods may or may not allow one to use the extended approach.
In addition to the question of whether the extended approach can be applied under a chosen version of the Bayesian paradigm, there is a question of whether it should be used. The major concerns surround our inability to check aspects of the model for future data-our inability to check the form of Y | (x, w) when w is unavailable and our inability to check the form of w | x-and our inability to consistently estimate the distribution of w | x as future data accrue. This last implies that, even as the future data set size tends to ∞, there will always be some uncertainty about the value of observing w.
The robustness to priors is also an issue when comparing approaches. The conditional distribution w | x is an additional (subjective) prior. This specification may or may not be correct, bringing additional sensitivity to the analysis. If it is based on scientific theory, it helps us to obtain accurate prediction at a lower cost. However, especially in the are of social science, it is unstable due to, for example, the advance of technology or the change of laws. The unstability affects the robustness of analysis to the priors.
Another aspect on the extended approach will be discussed in the next section.

Economic variable selection 4.1 Loss
When the data are observed, we are able to estimate the loss. Let y i be the response to be predicted for case i (i = 1, . . . , n), and let (x i , w i ) be the purchased and unpurchased predictors for case i, respectively. For each model γ, The uncertainties about models and parameters are estimated by the posterior distributions of models and parameters, which are associated with densities π (M γ | D γ ) and π (φ γ | D γ ), respectively. Then, the loss is estimated as where (ỹ,x γ ) is the new response and predictors for the subset γ and h(·) is an action to be chosen. Under the squared-error loss, the best action is the posterior conditional expectation E(Ỹ |x γ , D γ ).
When the new data are not available, the cross-validated loss is an alternative. The data are split into two parts: the training and validation data. Then, the conditional expectation and the posterior distributions are estimated based on the training data, and by using the validation data in place of the new data, we have the estimated loss. When the data are divided into several groups, this process is repeated by treating one of them as the validation and remainings as the training. The cross-validated loss is the average of these losses. In Section 5, the squared predictive loss is used as the adequacy measure and is estimated by the 10-fold cross validation.

Normal linear model with g prior
The framework described so far is general enough. This subsection specifies the statistical model and priors to be used to illustrate our methodology with real datasets in Section 5.
The statistical model is specified as the normal linear model. For γ ∈ Γ and i = 1, . . . , n, where the error term i independently and identically follows the normal distribution with mean 0 and variance σ 2 , i.e., i ∼ N (0, σ 2 ). Model parameters are φ γ = (β 0 , β 1,γ , σ 2 ) and predictors are standardized. The subscript for β 0 and σ 2 is suppressed because they are commonly used in all models.
The following prior distributions are assumed: where g and p are known constants and γ j is the indicator if the j-th predictor is in the model γ. The prior for slope coefficients (β 1,γ ) is called the g prior (see Zellner (1986) and Zellner and Siow (1980)) The constant g is set equal to the number of observations n, which is recommended by Fernández et al. (2001) when the number of observations is greater than the squared number of predictors.
It is possible to assume a hyperprior on g. One of the most common choices is the hyper-g distribution proposed by Liang et al. (2008). The benchmark prior by Ley and Steel (2012)  We tried the hyper-g prior with datasets used in the illustrative examples in Section 5, obtaining similar results to those without it. Thus, this study does not utilize a hyperprior on g.
Due to the lack of knowledge about models, we set p = 0.5, leading to the uniform prior over models. Other specifications of the prior model probability is found in, e.g., Steel (2017).
The (marginal) posterior of (β 0 , β 1,γ ) is a generalized multivariate t distribution, details of which are given in Appendix A (see Eq. (16)). The expected squared-loss loss conditional on the model γ is derived as Eq. (17)  Proof. Pick a model γ with m predictors (m ≤ k). Its log marginal likelihood is given by where R 2 is the coefficient of determinant. To this model, q predictors are augmented.
Then, the log marginal likelihood becomes where ∆ is the increase in the coefficient of determinant by the augmentation. Due to the orthogonality condition, the increase does not depend on which q predictors to augment.

The difference of two log marginal likelihoods is
Next, consider two models, models b and s, both of which have m predictors. Suppose the former has a higher R 2 than the latter, i.e., R 2 b > R 2 s . For these models, we have differences of log marginal likelihoods, and the difference of them is for ∆ > 0.
In terms of Bayes factors, log(BF bs+q ) − log(BF bs ) > 0, where log(BF bs+q ) = m(γ b+q ) − m(γ s+q ) and log(BF bs ) = m(γ b ) − m(γ s ). Therefore, as long as the prior model probability depends only on model size, posterior model probabilities for higher R 2 models are larger in the extended approach than those in the restricted approach.

Ozone dataset
The first dataset is originally analyzed by Breiman and Friedman (1985) to develop a model between the daily ozone concentration level and meteorological variables in Los Angeles.
We use the data provided by the R package 'bfp.' The number of observations is 330.
The response is the log daily ozone concentration level in 1976 measured at Upland, California. There are 10 possible predictors: (1) 500-millibar-pressure height, (2) wind speed, (3) relative humidity, (4) temperature at Sandberg, (5) inversion base height, (6) binary variable that is set one if the inversion base height is 5,000, (7) pressure gradient from Los Angeles International Airport to Daggett, (8) inversion base temperature, (9) square root of visibility, and (10)  x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 (a) All combinations. x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 (b) Top 128. is the (estimated) squared predictive loss plot in ascending order and the lower part is the map of corresponding combinations. Vertical solid lines on the upper parts will be explained later. Each column of the lower part represents a combination of predictors. When a cell of a column is filled by black, the predictor labeled on the y-axis is included. The marginal posterior probability that the coefficient is nonzero is expressed by the brightness of the cell. If it is close to one (zero), the cell becomes black (white). See Clyde (2003) for the marginal posterior nonzero probability.
The least-loss combination is (x 3 , x 4 , x 5 , x 9 , x 10 ). Among them, x 9 is less relevant in terms of its marginal posterior nonzero probability (about 0.66). Thus, the cell corresponding to x 9 is colored to be gray. Compared with the selection by Breiman and Friedman (1985), we choose x 3 instead of x 7 .
To focus on combinations that yield smaller predictive losses, the left panel is magnified to the right by picking up the top 128 combinations. In this panel, x 4 is always included in the top 128 models in terms of its nonzero probability. Among other variables, x 10 is in the models with smaller predictive losses.
Two special models are considered: the intercept-only model and the model that include all predictors. Their respective position is denoted by the vertical solid lines labeled by N and F in Figure 1. The former yields the high predictive loss, although it is not the worst (the fourth from the worst). On the other hand, the latter performs much better. This loss is achieved when we use the usual Bayesian model averaging. Its predictive performance is closer to the best (see the predictive loss plot of the left panel). However, there are more than two hundred combinations that yield less predictive losses and include less predictors.
Next, two cost structures are considered. The first one is the uniform cost structure, where all predictors are set at the same price. That is, when a model includes k predictors, the total cost is c · k, where c is the price. This structure is used when a decision maker has no information about the cost of predictors.
When the uniform cost structure is applied, selected predictors are shown by the left panel of Figure 2. Each column is the least-loss combination for a fixed c, which is indicated by the horizontal axis. Similar to the previous plots, the brightness represents the marginal posterior probability that the corresponding coefficient is nonzero. As c increases, less predictors are included in the model. When c is sufficiently high, the optimal model is the one with no predictors.
It is reasonable to consider that a decision is made with some knowledge about pre-x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 Cost (a) Uniform cost structure.
x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2  dictors. A possible decision maker for this dataset is a researcher who is interested in the global warming. As a part of his or her interest, the researcher would like to predict the ozone level. He or she probably knows the cost of predictors. One reasonable cost structure for the researcher is cost per predictor. Because (x 2 , x 3 , x 4 , x 9 , x 10 ) are often reported on regular weather news, it is natural to assume their costs are zero, while remaining predictors require positive prices. To simplify the structure, we assume each of them requires the constant price c. The results are shown in the right panel of Figure 2. As c increases, the optimal model is the one without x 5 because of its higher cost.

Diabetes dataset
Next dataset is the diabetes data, which are used in Efron et al. (2004)'s study and are provided through Professor Trevor Hastie's webpage. The data are used to predict the progression of the disease one year ahead of the baseline when predictors related to patients are collected. In this dataset, 442 observations are included.
The response is the log of diabetes progression measure. Ten possible predictors are included: (1) age, (2) sex, (3) body mass index, (4) blood pressure, and 6 blood serum measures. Again, predictors are standardized.
Results without cost are summarized by Figure 3. The least-loss combination is (x 3 , x 5 , x 9 ). x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 (a) All combinations. x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 (b) Top 128. From the top 128 combinations, x 3 is useful to predict the disease progression because it is always included in the combinations. Among 6 blood serum measures, x 9 is useful as well because it is almost always included in the models. Efron et al. (2004) applied the least angle regression and they find that variables comes into the active set in the order of x 3 , x 9 , x 4 , x 7 , where x 4 and x 7 are selected in combinations with higher predictive losses in our results.
The performance of two special models are examined. The intercept-only model is the fourth from the worst. The model that include all predictors does not perform well in this dataset. This dataset suggests that other combinations that include less predictors would be better to predict the disease progression.
Two cost structures are specified. The first one is the uniform cost structure and its results are on the left panel of Figure 4. As c (the uniform price) increases, the number of x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 0 0.1 0.2 Cost (a) Uniform cost structure.
x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 0 0.1 0.2 Cost (b) Cost per model. predictors in the optimal model decreases. The optimal model with sufficiently large price is the one only with the intercept.
A possible decision maker for this dataset is a person who is at the risk of diabetes. If he or she considers it low, the cost of blood test is expensive. On the other hand, if he or she considers it high, it becomes cheap. To simplify this decision problem, the constant cost c is introduced if either of 6 blood serum measures is included in the model. Other predictors are assumed to be free of charge.
The results are shown on the right panel of Figure 4. As the price for the blood test increases, (x 5 , x 9 ) are excluded in the optimal model because they become more expensive.
When it is sufficiently high, (x 5 , x 9 ) are replaced by x 4 , and (x 3 , x 4 ) are selected to predict the progression of the diabetes. For a person who is at the low risk of diabetes, these predictors are enough for the purpose.

Wage dataset
The last dataset focuses on how wage is determined by attributes of workers, such as the education level and the ability.
The dataset to be used in the analysis is the one taken from the National Longitudinal Survey of Youth and is the panel data from 1979 to 1993. This is analyzed by Koop and Tobias (2004) and is provided from the Journal of Applied Econometrics data archive.
The response is the log of hourly wage for white males. Koop and Tobias (2004) excluded observations who are at the age of less than 16 years or who report small wages, short working hours, or inappropriate education years. There are 7 possible predictors: (1) education in years, (2) potential experience in years (age − years of education − 5), (3) the ability measure ranging from about −4 to about 2, constructed on 10 component tests of the Armed Services Vocational Aptitude Battery, (4) mother's education in years, (5) father's education in years, (6) binary variable for broken home until the age of 14, and (7) number of siblings. The response and the first two variables are time variant, while the remaining five are time invariant. More details of this dataset are given in Section 4 of Koop and Tobias (2004). All predictors are standardized.
The least-loss combination for each wave is aligned in Figure 5. The ability measure x 7 x 6 x 5 x 4 x 3 x 2 x 1 1 5 10 15 Wave Figure 5: Least-loss combinations (wage).
(x 3 ) comes into the model after the fourth wave. A possible reason is as follows. For the first three years, companies mainly set wages by the education level (x 1 ) and the experience (x 2 ) because the ability is unknown. It will be turned out as working together. After about three years, companies start to use its information to set wages more accurately.
A decision problem in this dataset is when to purchase the ability measure as a manager of a company. When it is free of charge, purchasing at the beginning of t-th wave yields the prediction loss as t−1 s=1 l s + 15 s=t l * s , where l * s and l s are the least losses with and without purchasing the ability measure, respectively.
The left panel of Figure 6 plots this loss changing when to purchase. In this panel, the  symbol N represents no purchase for all waves. The minimum loss is reached at the either beginning of the first four waves, represented by the black dots.
The decision changes when cost is introduced. There are two kinds of cost in this problem: the discount factor (δ) and the price of the ability measure (c). Because the present and future utilities/costs are not equivalent, the discount factor is introduced to evaluate the future in terms of the present value. If the ability measure was purchased at the beginning of the t-th wave, the prediction loss with cost adjusted by the discount factor would be The loss is discounted because it is interpreted as the utility in the statistical decision problem or because it is measured by dollars so that it is additive to the cost.
The decision with cost is shown in the right panel of Figure 6. Each panel is for a fixed discount factor and ten different prices of the ability measure, that are ranging from 0.01 to 0.2. A gray line is the plot of the loss described above for a fixed price and discount factor, and a dot (dots) indicates the minimum.
As the price gets higher, the loss increases. When we see the top left panel as an example, the gray line moves up along the y-axis as the price increases. Thus, in this case, the optimal decision is around the beginning of the fourth to sixth wave. With the positive discount factor, it moves to a later wave or no purchase as the price increases. See the top right panel, for example.
As the discount factor becomes larger, the loss plot becomes downward because the decision maker values the present more than the future. Then, the optimal decision tends to purchase the measure at a later wave even if it improves the loss. When the discount factor and price are sufficiently large, the optimal decision is no purchase. It is reasonable that the manager of a company purchases the ability measure later or decides no purchase of it when it is expensive and/or the discount factor is large.

Discussion
The variable selection problem depends on the person who chooses predictors. The additive cost is a simple way to incorporate this aspect. Further, given a set of purchased predictors, there is uncertainty about models as well as parameters. The Bayesian model averaging is a cohesive approach to address it. In a broader view, it is considered to be the restricted approach. The extended approach is available when the subjective approach is allowed or we have information about unpurchased predictors given purchased ones.
The optimal experimental design is a research area related to our work. When the squared-error loss is interpreted as the utility function, the proposed method can be used for the optimal experimental design where predictive performance is of interest (see, e.g., Chaloner and Verdinelli (1995) for a survey of its Bayesian version).
Finally, a computational limit is noted. The method is computationally feasible when the number of predictors is moderate. However, for example, the growth model usually includes more than fifty predictors (sixty seven in Sala-i-Martin et al. (2004)). In this case, it is necessary to remove weak predictors before proceeding to apply our method or use a specific algorithm proposed by Papaspiliopoulos and Rossell (2017) for example.

A Posterior, loss, and marginal likelihood
This section derives the posterior, the loss, and the marginal likelihood under the normal linear regression model specified in Subsection 4.2. The subscript γ is suppressed in this section to simplify notations.