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## Soccsld.dvi

⊲ 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.

Source: http://econ.as.nyu.edu/docs/IO/22512/Gallant_Slides_02172012.pdf

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