⊲ accommodates a nondifferentiable, non-
⊲ with (mixed) continuous or discrete state
⊲ using a locally linear value function
⊲ Entry and exit from industry, technol-
• Use sequential importance sampling (parti-
⊲ to integrate unobserved variables out of
⊲ to estimate ex-post trajectory of unob-
⊲ Stationary distribution of MCMC chain
⊲ Because we prove the computed likeli-
⊲ Efficient, number of required particles is
⊲ Monopolistic competition, twenty play-
• There are i = 1,. , I, firms that are iden-
• Firms maximize PDV of profits over t,. , ∞
∗ Capacity constraint: increased costs.
• Each period t a market opens and firms
⊲ If enter Ai,t = 1, else Ai,t = 0.
⊲ Firms know each other's revenue and
• Number of firms in the market at time t, is
• Gross revenue Rt is exogenously determined.
• A firm's payoff is Rt/Nt − Ci,t where Ci,t is
• Costs are endogenous to past entry deci-
⊲ ci,u,t = µc + ρc (ci,u,t−1 − µc) + σceit
V (Ct, Rt) = (V1(C1t, C−1t, Rt),. , VI(CIt, C−It, Rt))
⊲ ci,k,t = ρa ci,k,t−1 + κaAi,t−1
- V (ct, rt) is approximated by a local linear function.
- The integral is computed by Gauss-Hermite quadra-ture.
• Coordination game: If multiple equilibria
(rare), the lowest cost firms are the en-trants.
• The value function V is approximated as
s = (cu,1,. , cu,I, r, ck,1,. , ck,I).
−i,t, Ci,t, C−i,t, Rt) ≥ Vi(Ai,t, AE
Each hypercube of the grid is indexedits centroid K, called its key. The local
linear approximation over the Kth hy-percube is VK(s) = bK + (BK)s.
⊲ For a three player game VK is 3 × 1, bK
is 3 × 1 , BK is 3 × 7, and s is 7 × 1.
−i,t+1, Ci,t+1, C−i,t+1, Rt+1)|Ai,t, A−i,t, Ci,t, C−i,t, Rt]
• The local approximator is determined at
is the choice-specific payoff function.
Complete information: Ct, Rt known implies AE
−i,t+1, Ci,t+1, C−i,t+1, Rt+1) = Vi(Ci,t+1, C−i,t+1, Rt+1)
efficients bK and BK by regressing {Vj} on{sj}. Continue until bK and BK stabilize.
⊲ Usually only 6 hypercubes are visited.
⊲ ci,u,t = µc + ρc (ci,u,t−1 − µc) + σceit
⊲ ci,k,t = ρa ci,k,t−1 + κaAi,t−1
• Parameters: θ = (µc, ρc, σc, µr, σr, ρa, κa, β, pa)
⊲ Firms take outcome uncertainty into ac-
⊲ Bellman equations modified to include
• We can draw from p(x1t | at−1, xt−1, θ) and
where x1t is not observed and x2t is observed.
The observation (or measurement) density is
p(xt | at−1, xt−1, θ) and discarding x2t.
• There is an analytic expression or algorithm
to compute p(at | xt, θ), p(xt | at−1, xt−1, θ),
• If evaluating or drawing from p(x1t|at−1, xt−1, θ)
is difficult some other importance samplercan be substituted.
2. Given the parameter value and the seed,
compute an unbiased estimator of the in-tegrated likelihood.
• Compute by averaging a likelihood that
3. Use the estimate of the integrated likeli-
Main point:Deliberately put Monte Carlo jitter into theparticle filter.
t | xt, θ) p(xt | at−1, xt−1, θ)
p(x1,0 | θ) using s as the initial seed and
p(at|xt, θ)p(xt|at−1, xt−1, θ)dx1t
(b) If p(at, x2t | x1,t−1, θ) is available, then
• Integrate by averaging sequentially over
nated draws for fixed k that start at times and end at time t are denoted
(f) Note the convention: Particles with un-
After resampling the particles have equalweights 1 and are denoted by {x(k)
(a) An unbiased estimate of the likelihood
and s′ is the last seed returned in Step 2e.
• An unbiased estimate of the likelihood is
• In the Bayesian paradigm, θ and {at, xt}∞
are defined on a common probability space.
• The elements of at and xt may be either
real or discrete. For z a vector with some
coordinates real and the others discrete,
Lebesgue ordered to define an integral of
• Particle filters are implemented by drawing
independent uniform random variables u(k)
and then evaluating a function∗ of the form
k = 1,. , N. Denote integration with re-
∗E.g., a conditional probability integral transformation.
for integrable g(x1t), we seek to generate draws ˜
• Put 1 = g(x1,0:t) = g(x1,0:t, x1,t+1).
g(x1,0:t, x1,t+1) dP (x1,0:t, x1,t+1|Ft+1)
• Algebra to express the numerator of ˜
• Show that resampling does not affect the
result as long as scale is preserved.
• Use a telescoping argument to show that
weights can be normalized to sum to oneat a certain point in the algorithm.
• Three firms, time increment one year.
⊲ µc and µr imply 30% profit margin, per-
⊲ κa is a 20% hit to margin with ρa at 6
⊲ σc and σr chosen to prevent monopoly
⊲ Outcome uncertainty 1 − pa is 5% (from
θ = (µc, ρc, σc, µr, σr, ρa, κa, β, pa)
= (9.7, 0.9, 0.1, 10.0, 2.0, 0.5, 0.2, 0.83, 0.95)
T0 = 160, sm: T = 40, md: T = 120, lg: T = 360
the columns labeled "lg" would not givemisleading results in an application.
1. Fit with blind importance sampler, and
• In smaller sample sizes the specification er-
2. Fit with adaptive importance sampler, and
ror caused by fitting the boundedly ratio-
nal model to data generated by the fullyrational model can be serious:
3. Fit with adaptive importance sampler, and
columns "sm" and "md" in Tables 3 and 4.
The saving in computational time is about10% relative to the fully rational modelso there seems to be no point to usingthe boundedly rational model unless that iswhat firms are actually doing, which theyare not in this instance.
• Constraining β is beneficial: compare Fig-
bimodality of the marginal posterior dis-
tribution of σr and pushes all histogramscloser to unimodality.
small savings in computational cost: com-
• Improvements to the particle filter are help-
ful. In particular, an adaptive importance
sampler is better than a blind importance
pare Figures 3 and 4. Systematic resam-pling is better than multinomial resam-pling; compare Tables 5 and 6. Histogram of mu_c Histogram of rho_c Histogram of sigma_c Histogram of mu_r Histogram of sigma_r Histogram of rho_a Histogram of kappa_a Histogram of beta Histogram of p_a Firm 1's log unobserved cost Histogram of mu_c Histogram of rho_c Firm 2's log unobserved cost Histogram of sigma_c Histogram of mu_r Histogram of sigma_r Firm 3's log unobserved cost Histogram of rho_a Histogram of kappa_a Histogram of p_a
Circles indicate entry. Dashed line is true unobserved cost.
The solid line is the average of β constrained estimates over
all MCMC repetitions, with a stride of 25. The dotted line is± 1.96 standard deviations about solid line. The sum of the
norms of the difference between the solid and dashed linesis 0.186146.
β Constrained, Adaptive Sampler, Md. Firm 1's log unobserved cost Firm 2's log unobserved cost Firm 3's log unobserved cost
Circles indicate entry. Dashed line is true unobserved cost. The solid line is the average of β constrained estimates over
all MCMC repetitions, with a stride of 25. The dotted line is± 1.96 standard deviations about solid line. The sum of the
norms of the difference between the solid and dashed linesis 0.169411.

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