CAPITULO III EL PERSONAL
3.5 Impuesto Sobre la Renta de los Trabajadores
This paper exploits the Nested-Pseudo Likelihood Algorithm developed by Aguirregabiria and Mira (2002, 2007). Below, I describe some details of the estimation procedure in this paper. The estimation consists of the following steps:
1. Estimation of the demand equation 2. Construction of the state space:
• construction of the demand conditions using the result of the demand equation.
• the discretization of the state variables 3. Estimation of the initial choice probabilities
4. Estimation of the laws of the evolutions of the state variables, and . 5. Estimation of the dynamic parameter
The details of the estimation from 1 to 4 are discussed in the main text. I talk about the step 5 and 6. In the actual practice, I modified the procedure described in Aguirregabiria(2001) ”A Gauss Program for the estimation of discrete choice dynamic programming models using a Nested Pseudo Likelihood Algorithm.”
Step 5: The estimation of transition probability matrices of the state variables
I construct the transition probability matrices for fd(dit|dit−1) and frp(rpt|rpt−1) as follows. The transition probability matrix for rp is used in the monopolistic competition model. For example, the stochas-tic process of the demand condition for brand i is specified as follows:
dit= δd0+ δd1dit−1+ dit (A3-1)
where dit and dit−1 are continuous demand conditions; δd0 and δd1 are the coefficients; and dit follows an iid distribution function fd
it. The process of the rival price is specified in the analogous manner. The coefficients of the above process are estimated using OLS. Then, using the Kernel density estimation, I derive the distribution of the residual non-parametrically. I construct the transition probability matrix of dt
counting the frequency of the realization of each pair of ditand dit−1. Step 6: The estimation of the menu-cost parameter
According to Aguirregabiria and Mira(2007), I derive an alternative presentation of value functions and conditional choice probabilities, which are used in the pseudo-likelihood estimation of a menu-cost parameter.
Let P∗ be a matrix of equilibrium probabilities, which are best response probabilities, and ViP∗ be the corresponding value functions of brand i. Using P∗and ViP∗, I can rewrite the Bellman equation (7) as
ViP∗(x) =X on x. 2 In vector form, equation (A3-2) is
ViP∗=X dimension M , which is the size of the state space. FP∗ is a matrix of transition probabilities of fP∗(x0|x).
Under the condition β < 1, the value function given P∗ can be obtained as the solution of the following linear equation:
(I − βFP∗)ViP∗=X
j∈J
Pi∗(j)[ΠPi∗(j) + ePi∗(j)], (A3-4)
where I is an identity matrix with dimension M . Denote the mapping for the solution of equation (A3-4) as Γi(x; P∗). For an arbitrary set of probabilities P , the mapping operator Γi(x; P ) gives the values for brand i when all the brands behave according to P . Note that this mapping is constructed given the conditional choice probabilities of brand i as well as those of its rival brand. Using this mapping Γ, instead of ViP in equation (A3-4), I define a mapping Ψ to calculate the expected value for brand i to choose action ai for P :
Ψi(j|x) =
Next, the pseudo-likelihood function to estimate the menu cost is derived. For convenience, define the following notations: an expected price of a competing brand under its conditional choice probability P , pP−it=P
j−iP (j−i| xt)p−it for given xt. Given the estimated coefficients of the demand equation and the constructed demand conditions ˆdit, I set up the expected one-period profit associated with action a as
ΠbPi (j, xt) = (wjit− cit)( ˆdit− bb0(pjit) + bb1ln(pP−it)) + ( bb2pist+ bb3p−ist) × br − γI{j = 1}. (A3-6) For exposition, denote bΠPi (j, xt) = zitPθ, where zitjP = {(witj − cit)(dt− ˆb0(pjit) + ˆb1(pP−it)( bb2pist+ bb3p−ist) × br), −I{j = 1}} and θ = {1, γ}. Let FP be the transition probability matrix representing all the transition processes of the state variables x under the conditional choice probabilities P , and ePi∗(j) be a vector of the expectation of εji conditional on x.3
The empirical counterparts of the value functions and the best response probabilities are derived according to the mapping expression by Aguirregabiria and Mira(2007). Let Γi(P ) denote the mapping operator of the value function in vector form given conditional choice probabilities P , and Ψ(j | x) be the operator representing the best response probabilities given Γi(P ). Γi(P ) can be written as Γi(P ) = ZiPθ+τiP, where ZiP = (I − βFP)−1P
jPi∗(j)Πi(j) and τiP = (I − βFP)−1P
j∈JePi∗(j), where the value of discount factor is assumed to be known a priori and fixed at 0.99. Assume that private information follows an i.i.d.
Type I Extreme Value distribution.Then, ePi (j) = Euler’s constant − ln(Pij), where Euler’s constant is about 0.577. The mapping of the best response probabilities Ψi given P is
Ψi(j) = exp{zjitθ + βFj(ZiPθ + τiP)}
P
jexp{zitjθ + βFj(ZiPθ + τiP)}. (A3-7) I construct a pseudo-likelihood function to estimate θ treating the conditional choice probability as nuisance parameters. Let Poand θo denote the true conditional choice probabilities and menu costs. Given the true conditional choice probabilities Po, the corresponding pseudo-log-likelihood function is
2 θo. The NPL estimator is obtained by the following procedure. I conduct the pseudo-maximum likelihood estimation of θ given a vector of initial values of conditional choice probabilities, P0, and then obtain the updated ˆP1using ˆθ1according to the mapping Ψ. I iterate this procedure for K ≥ 1 stages. In the estimation, the K-stage pseudo-log-likelihood is constructed as:
2
Letting ˆθK denote the structural parameter that maximizes equation (??) in the Kth stage, I can obtain the K-stage estimator of conditional choice probabilities:
PˆK= Ψ( ˆPK−1; ˆθK). (A3-10)
Under standard regularity conditions, it is consistent and asymptotically normal. Moreover, the estimator gains efficiency by repeating for K > 1 stages compared to the estimator without the iteration in terms of K. In practice, I conduct the estimation for K stage until ˆPK = ˆPK−1 or, equivalently, ˆθK = ˆθK−1 is obtained. The estimates converge fairly quickly (within 20 iterations). Note that the conditional expected profit except the menu-cost parameter consists of the conditional expected demand times price-cost margin in the U.S. dollars. Since the parameter of menu cost has the same unit as the conditional expected profit as specified in θ, the estimated γ is interpreted in the same unit.
3 FP = P
ji
P
j−iP (ji) ∗ P (j−i) ∗ (Fip⊗ F−ip ⊗ Fid⊗ F−id ⊗ Fic⊗ F−ic ), where * represents the element-by-element product, ⊗ represents the Kronecker product, and Fiprepresents the matrix of the transition probability fip, respectively.
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Table 1: Summary statistics of brands
Brand nob Size of a box Market share Mean price Mean quantity sold
unit oz % $/oz oz
Keebler 7333 15 18.0 0.17 118
Nabisco 7485 16 34.6 0.16 234
Sarelno 7418 16 16.9 0.15 126
Dominick’s 7340 16 30.4 0.12 280
1. Market shares are those of revenue.
2. Prices are nominal.
3. The observations are those with a positive purchase. The statistics are calculated before list-wise deletion for the estimation.
Table 2: Frequency of nominal price changes
Brand nob Price Changes Downward Upward Yearly change
unit % % % times per year
Keebler 7057 31.8 16.2 15.6 16.5
Nabisco 7330 33.9 17 15.6 17.6
Sarelno 7193 22.5 11.4 11.1 11.7
Dominick’s 7142 26.8 14.1 12.8 13.9
average 32.85 16.6 15.6 17.05
(Nabisco & Keebler)
average 24.65 12.75 11.95 12.8
(Sarelno & Dominick’s)
1. Yearly change is the average number of price changes per year (52 weeks).
2. The observations are those with a positive purchase and a lagged value. The statistics are calculated before list-wise deletion for the estimation.
Table 3: Summary statistics of bonus
nob frequency mean
length
unit % week
Keebler 7579 28 2.3
Nabisco 7579 21 2.1
1. The observations are for all the weeks in the sample.
Table 4: Summary statistics of cost ($U.S. per oz) mean std.dev min max cost (nominal) 0.04 0.003 0.04 0.05 cost (real) 0.04 0.001 0.04 0.041
Table5:DemandEstimationResults:OLSandIVwithoutliers VariableOLS1OLS2OLS3IV1IV2IV3IV4 price-32.43(1.03)-29.23(1.62)-29.82(1.39)-39.02(2.16)-38.94(2.70)-39.72(3.18)-39.49(6.44) rp5.78(1.00)8.36(1.82)8.36(1.49)2.97(1.58)13.87(2.43)14.05(2.41)14.12(3.41) sdp0.33(1.42)0.38(1.74)0.43(1.42)18.18(7.17)15.05(7.33)13.72(7.67)5.77(10.33) price×br-7.52(2.41)-7.84(2.01)2.32(4.96)1.91(4.69)7.39(7.17) rp×br-5.21(2.41)-4.59(1.97)-14.95(3.32)-15.47(3.30)-13.06(5.45) cc3.67(0.35)3.68(4.65)3.72(0.35)3.65(0.39)3.69(0.38)3.75(0.38)3.65(0.42) br6.79(0.24)25.71(0.91)25.16(3.88)6.62(0.26)25.27(9.54)26.61(8.88)14.39(17.96) bonus1.68(0.47)1.51(0.74)1.54(4.25) bonusduration-0.043(0.10)-0.16(0.09)0.73(1.11) bonusduration2-10.29(1.72)-11.18(1.95)-6.08(14.13) constant33.35(3.02)24.16(3.55)25.82(3.65)27.51(9.39)15.65(9.63)20.09(13.59)26.85(23.86) N14024140241402413120131201312012308 R2 0.570.570.570.560.560.560.56 J-stat(p-value)3.91(0.27)2.67(0.45)1.80(0.61)6.08(0.11) elasticity(average)-2.80-2.52-2.57-3.37-3.36-3.43-3.41 elasticity(Keebler)-4.34-3.91-3.99-5.22-5.21-5.31-5.28 elasticity(Nabisco)-2.04-2.31-2.37-2.45-2.30-2.38-2.02 meandemandcondition53.3044.2346.0046.8134.4652.8352.65 1.Theregressionsincludeadummyvariableforoutliers,store-dummy,andtime-dummyvariables(monthandyear). 2.Pricevariablesarereal-valued.TheunitistheU.S.dollarsper10ounces. 3.Thedependantvariableislogofquantitysoldin10ounces. 4.TheendogenousvariablesinIV1-IV4areprice,rp,sdp,price×br,andrp×br.IV4additionallytreatsbonus,bonusduration,bonusduration2 asendogenousvariables.TheexcludedinstrumentsareAAC,rivalAAC,AACofSalernoandDominick’s,thelaggedvariablesofthesethreeAAC, thecrosstermofAACandbranddummy,thecrosstermofrivalAACandbranddummy.IV4includesthesecondorderlaggedAACvariablesas theexcludedinstrumentsaswellasthevariablesusedforIV1-IV4. 5.Standarderrorsareinparenthesis.Theyareheteroscedasticityandauto-correlationrobust.
Table 6: Summary statistics of state variables nob mean std.dev price (per 10 ounces) 3678 1.49 0.12 demand condition (IV4) 3694 52.65 4.89 cost (per 10 oz) 3678 0.39 0.01
Table 7: Mean statistics of the Wholesale Price
Keebler Nabisco wholesale price ($U.S. per 10 ounces, deflated) 1.07 1.10
frequency of large price change (%) 27 25
nob 1839 1839
Table 8: Estimated menu costs
monopolistic competition duopoly ˆ
γ 4.53 1.96
(0.10) (0.06)
log-likelihood -945 -503
nob 3528 3528
1. The estimated results are based on the results of IV 4.
The size of the state space is 1800 (np=5, nd=6, nc=1, nbr=2).
2. The standard errors are inside parenthesis and
based on 5000 non-parametric bootstrapping re-samples.
Table 9: Estimated menu costs by different results of the demand estimation
OLS1 OLS2 OLS3 IV1 IV2 IV3 IV4
ˆ
γ 1.71 1.64 1.73 1.71 1.66 1.96 1.96
s.e. (0.05) (0.05) (0.05) (0.12) (0.10) (0.06) (0.06) log-likelihood -478 -449 -433 -723 -677 -506 -503
nob 3678 3678 3678 3528 3528 3528 3528
1. The size of the state space is 1800.
2. The standard errors are inside parenthesis and
based on 5000 non-parametric bootstrapping re-samples.
Table 10: Estimated menu costs and the numbers of grids
nd\np 2 3 4 5
2 1.37 2.40 -
-3 1.62 1.82 2.50 1.78 4 1.73 1.54 1.99 2.43 5 1.80 1.38 1.88 2.12 6 1.83 1.27 1.81 1.96
1. The estimated results are those of the duopoly model. The specification of the demand model is IV4.
2. ’np’ stands for the number of grids of price and rp. ’nd’ stands for the number of grids of the demand condition, d.
3. No convergence was achieved for -.
Table 11: Comparison of estimated menu costs with previous studies size % in revenues
this study: ˆγ 1.96 18
Levy et al. 0.52 0.7
Slade 2.55 5.11 §
Aguirregabiria 1.45 † 0.7
Nakamura and Zerom 7000 0.23
1. §The value is calculated from Table IA and VB as the share-weighted average.
2. † The value is calculated from Table 6. The reported value in this table is the result from the specification allowing for asymmetric menu costs. The result without asymmetry is close to this value.
Figure 1: Prices of the Four Brands of Standard Graham Crackers
0 50 100 150 200 250 300 350 400
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
week
$U.S. per ounce
Salerno Keebler Dominick’s Nabisco
37
Figure 2: Choice Probabilities of No-Price Change at γ=1.96
MC stands for the monopolistic-competition model.
38
Figure 3: Price Rigidity and Own Demand Elasticity
1. MC stands for the monopolistic-competition model.
2. The coefficient on the rival price is set to be 1.
39
Figure 4: Price Rigidity and Strategic Complementarity
1. MC stands for the monopolistic-competition model.
2. The coefficient on the own price is set to be -30.
40
Figure 5: Comparison with the model with myopic agents
0.110.120.130.140.150.160.170.18 0.19
0.2
1. MC stands for the monopolistic-competition model.
2. The coefficient on the own price is -30.
3. The coefficient on the rival price is 1.
41