VNV="Virtually No Variance" model, with nc = 20.0 ; LV="Low Variance" model, with nc = 2.0. t {x,y} indicates that only sample observations between and including years x and y in the 80-year sequence of simulated prices are included in the estimation. Standard errors are given in parentheses. Additional explanation about this Table is found in the text. Starting values for all estimations: ft = 0.95, nc = 10.0, e1 = 0.20.

VNV="Virtually No Variance" model, with nc = 20.0 ; LV="Low Variance" model, with nc = 2.0. t {x,y} indicates that only sample observations between and including years x and y in the 80-year sequence of simulated prices are included in the estimation. Standard errors are given in parentheses. Additional explanation about this Table is found in the text. Starting values for all estimations: ft = 0.95, nc = 10.0, e1 = 0.20.

expected harvest age falls with an increase in the discount rate (decrease in the discount factor).

It should be emphasized that the low estimated discount rate in Model 5 is not a fluke associated with the particular starting values used in estimation. In a search across a range of starting values, including the parameters of the true model, we could not find estimates generating a higher likelihood value for the sample than those presented in Model 5. It should also be emphasized that the deep probability trough observed in Figure 15.7 occurs outside the range of the simulated data. Of the 8,621 harvests observed in the data, 8,561 (99.3%) occur at stand age 10 or less, and all harvests occur at a stand age less than 20. In other words, for very few observations is the stand age greater than 10, and for no observations is it greater than 20. Perusal of Figure 15.7 shows, then, that in the range of the data the harvest probability of Model 5 is virtually the same as for the true model. This teaches an old and familiar lesson, one certainly not unique to the estimation of DDP's: in the absence of a good range in the data, identification of a model can be difficult.

4. Comments on Reduced Form (Static) Estimation of Discrete Dynamic Decision Problems

Given the difficulty of structural estimation of a discrete DDP, it is reasonable to question whether the effort is worth the gain. In particular, why not

Probability Difference

Probability Difference

Figure 15.7. Harvest Probability Difference between the True Model and Model 5

Figure 15.7. Harvest Probability Difference between the True Model and Model 5

simply estimate a reduced-form version of the problem? Inspection of the timber harvesting decision (3.8) makes clear that one can specify the decision problem as one in which the forest owner harvests trees if:

where Aet is distributed logistically. This leads to a straightforward application of logistic maximum likelihood estimation. A similar reduced form can be used to approximate the optimal decision rule of any discrete DDP. Of course, the presence of the value functions in (3.8) argues for a flexible form in the approximation, and even then the quality of the approximation may be poor. Figures 15.8-15.10 presents harvest probabilities from first-, second-, and third-order estimation of (4.1) for the case where the true model is the LV model (as = 2.0), and so the figure against which the panels of Figures 15.8-15.10 are to be judged is Figure 15.3. Estimates are based on simulated data of the same size as used in Table 15.1 - namely, pooled time-series, cross-sectional data of length 80 years and width 500 forest stands. The structural counterpart in estimation is Model 4 (see Table 15.1), which generated harvest probabilities virtually identical to the true harvest probabilities (that is, it generates a probability surface that looks exactly like the probability surface of the true model presented in Figure 15.3, so we do not bother presenting the surface here). A comparison of Figure 15.8 and Figure 15.3 indicates that when F ( ) takes the simple linear form,

F = a0 + aip + a2a, the approximation to the harvest probability surface is very poor, and the probability of harvest decreases as the stand age increases. Figure 15.9 indicates that when F ( ) takes the quadratic form,

F = ao + a1p + a2a + a3p2 + a4a2 + a5p • a the approximation is considerably improved, and Figure 15.10 indicates that when F (•) takes the cubic "form,"

F = a0 + a1p + a2a + a3p2 + a4a2 + a5p • a +a6p3 + a7a3 + agp2 • a + a9p • a2, the approximation is excellent. Seemingly and not surprisingly, even a fairly low-order polynomial will do a good job of approximating fairly complex decision rules.

The conceptual weakness of reduced form estimation is the same for DDPs as it is for any economic model; it does a reasonably good job of describing the effect of various state variables on decision variables, but the estimation is otherwise devoid of economic content. It identifies the variables affecting a dynamic decision, but it provides no insight about the mechanism of the relationship. In the illustrative example presented here, reduced form estimation tells the analyst that timber price and stand age do affect the harvest decision, but it is silent about the discount rate used in the harvest decision, an important issue in the long debate about whether forest owners harvest timber too soon (and therefore require incentives to take the long view); it says nothing about the nontimber benefits of forestland (as embodied in e), a major issue in the allocation ofland across various uses; it says nothing about the overall value of forestland; and it says nothing about the price expectations of forest owners.5 A related issue is whether it is safe to assume that a decision problem is static. For our timber example, one might simply posit that in each period the forest owner harvests if the intra-period utility from harvesting is greater than

5In the structural models estimated in this chapter, price distribution parameters are fixed at their true values. In other estimations conducted in preparation of the chapter but not presented here, price distribution parameters were included in the set of parameters to be estimated. Generally results were excellent, though a bit sensitive to starting values. Provencher (1995a,1995b) estimates models of pulpwood harvesting in which the price process is autoregressive.

Figure 15.8. Harvest Probability Surface for Linear Reduced-Form (Logit) Model.

the utility of not harvesting. For instance, letting H(at) denote the money-metric utility derived from a standing forest at age at, and assuming that harvest occurs at the beginning of the period and H(0) = 0, harvest occurs if ptW(at) _ c + ei > H(at) + s°

^ ptW (at) _ c _ H (at) > Aet, which, under the same assumptions about the distribution of e used above, is the basis of a simple logistic regression. However, this approach can be problematic. For many problems—timber harvesting surely among them—it is not reasonable to assume away dynamic behavior. For problems where dynamic behavior is an open question, things are a bit complicated. Baerenklau and Provencher (2004) examine the issue of whether recreational anglers allocate a "fishing budget" over the course of a season. We estimate both a structural dynamic model and a reduced form static model of trip taking behavior and find significantly different welfare estimates across models. More troubling, we also demonstrate theoretical inconsistencies and identification problems with the static model when behavior is truly dynamic.

Figure 15.9. Harvest Probability Surface for Quadratic Reduced-Form (Logit) Model.

Furthermore, although dynamic models encompass static ones, actually testing for static behavior is problematical for the simple reason that if one specifies a sufficiently flexible form for the static utility function, the static model will provide an excellent fit to the data. Put another way, a reduced form model can be relabeled as a structural, albeit static, model, and as already demonstrated, reduced-form models can provide an excellent fit to the data. In our timber example, a sufficiently flexible static form requires making H ( ) a function of p as well as a, and this is difficult to justify. Such structural restrictions are ultimately necessary to test static versus dynamic models.

For many problems there is little a priori knowledge about the intraperiod benefit (utility, profit) function, and so it is difficult to distinguish static from dynamic behavior. For such problems, out-of-sample forecasting may shed light on whether behavior is static or dynamic, but this is a quite expensive diagnostic. Baerenklau and Provencher (2004) conduct such a test and find that in general a dynamic model does a better job of predicting out-of-sample trip-taking behavior than does a static model.

Figure 15.10. Harvest Probability Surface for Cubic Reduced-Form (Logit) Model.

5. The Future of Structural Estimation of Dynamic Decision Processes

For years, structural estimation of DDP's simply was not practical because of the substantial barrier posed by the computational requirements of estimation. Rapid advances in computational speed in the mid-1990s reduced this barrier considerably, yet the literature estimating DDP's is decidedly modest, and there is no evidence that it growing. This is partly because the econometric modeling is difficult to understand and implement; this chapter attempts to clarify the estimation method. Perhaps it also reflects two related objections.

First, estimation requires strong parametric assumptions about the decision problem generating the data. In the example considered in this chapter, we maintained the strong assumption that the forest owner knows the growth function of trees and, perhaps even more unlikely, that the forest owner knows the stochastic process generating timber prices. In the real world, such assumptions are complicated by the requirement that the analyst correctly specify the parametric form of various dynamic processes, where by the "correct" process we mean the process actually used by the decision-makers.

Second, the estimation maintains that agents are dynamic optimizers. In a working paper entitled, "Do People Behave According to Bellman's Principle of Optimality?", Rust (1994a) makes the important point that any data set of state variables and decision variables can be rationalized as the outcome of dynamically optimizing behavior. The issue becomes, then, whether the data generated can be rationalized by dynamically optimal behavior circumscribed by plausible parametric specifications of the intra-period benefit function and dynamic processes. The issue returns, in other words, to a variation of the first objection: Is the parametric specification of the dynamic model plausible in some theoretical or practical sense?

Given that a flexible static model can fit a data set as well as a dynamic one, and that dynamic optimization per se does not imply testable restrictions (and thus is not refutable), how might an analyst who suspects that behavior is dynamic proceed? In our view, estimation of DDP's should be informed by agent self-reporting about expectations of future states and the relative importance of the future in current decisions. So, for instance, if agents respond in surveys that future states are unimportant to their current decisions, it would seem difficult to justify a dynamic model. If agents report that they believe the best guess of prices tomorrow is the current price, a random walk model of price expectations (or at least, a specification of prices that allows a test for a random walk) is warranted.

Yet even as economists are uneasy about the strong parametric assumptions to be made in the estimation of DDP's, there exists a longstanding uneasiness in the profession about using agent self-reports to aid in the estimation of dynamic behavior. Manski (2004) argues that with regards to agent expectations, this is not much justified. The author states,

Economists have long been hostile to subjective data. Caution is prudent, but hostility is not warranted. The empirical evidence cited in this article shows that, by and large, persons respond informatively to questions eliciting probabilistic expectations for personally significant events. We have learned enough for me to recommend, with some confidence, that economists should abandon their antipathy to measurement of expectations. The unattractive alternative to measurement is to make unsubstantiated assumptions (pg. 42).

Although the role of self-reports in an analysis can be informal, giving the analyst a rough basis on which to choose a static vs. a dynamic model, to choose one parametric specification of dynamic processes over another, and so forth, a useful direction of future research is to incorporate survey data in the estimation problem itself, similar in principle to the way revealed- and stated-choice data are integrated in the estimation of random utility models. We anticipate that in the next ten years insights gained from experimental psychology, experimental economics, and survey research will provide the basis for richer and more accurate models of dynamic economic behavior.

Chapter 16

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