Estimating (S,s) rule regression models David Vincent Independent - PowerPoint PPT Presentation

The model Estimation Stata xtss command Numerical example Conclusion References Estimating (S,s) rule regression models David Vincent Independent 2019 London Stata Conference 5 September 2019

The model Estimation Stata xtss command Numerical example Conclusion References Contents The model 1 Estimation 2 Stata xtss command 3 Numerical example 4 Conclusion 5

The model Estimation Stata xtss command Numerical example Conclusion References Introduction There are many economic variables such as product prices, or firm employment levels that exhibit infrequent adjustments. These outcomes can occur when there are costs associated with making changes (e.g. menu costs), which lead agents to adopt an (S,s) decision rule. Such rules are characterized by a band of inaction, where agents tolerate some deviation from an optimal frictionless outcome, provided the deviation is not too large. This presentation describes a new command xtss for estimating state dependent (S,s) models, for panel data applications. This is based on Fougere et al. (2010) and Dhyne et al. (2011).

The model Estimation Stata xtss command Numerical example Conclusion References Literature Fougere et al. (2010) estimate a price rigidity model to assess the impacts of the minimum wage on prices in French restaurants. This is based on a flexible (S,s) model where the thresholds vary over time and across restaurants. Dhyne et al. (2011) use price observations on various goods in Belgium and France to study the importance of real and nominal rigidities in price adjustments. They find that asymmetry in price adjustments are caused by trends in cost and not asymmetry in the (S,s) bounds. Gautier and Saout (2015) use a similar specification to Fougere et al. (2010) to examine the speed at which refined oil prices are passed-through to gasoline prices.

The model Estimation Stata xtss command Numerical example Conclusion References The model Consider the case where the dependent variable is the price of product i = 1 , .., N in period t = 1 , .., T denoted p it . Letting p ∗ it denote the latent desired or frictionless price, the price decision rule is given by:  it − p it − 1 > c u if p ∗ p ∗  it it it − p it − 1 < − c d p it = p ∗ if p ∗ (1) it it  − c d it − p it − 1 ≤ c u if it ≤ p ∗ it . p it − 1 where c u it and − c d it are stochastic (S,s) bounds which differ across i and t and allow for asymmetric menu costs. Price equals the frictionless value when the difference it − p it − 1 is above c u it (price rise) or below − c d p ∗ it (price fall).

The model Estimation Stata xtss command Numerical example Conclusion References The model In the analysis of prices, the thresholds measure the extent to which changes are costly and represent nominal rigidity. Dhyne et al. (2011) assumes these are normality distributed with time-invariant means; however this leads to a non-zero probability of a price rise and a price fall. it − p it − 1 = 0 . 1, c u it = − 0 . 2 and − c d For example if p ∗ it = 0 . 2, the gap is above the upper bound but also below the negative of the lower bound. Such outcomes are inconsistent with the price decision rule where the thresholds must be non-negative.

The model Estimation Stata xtss command Numerical example Conclusion References The model To avoid this issue, I modify their model and allow the thresholds to have a normal distribution truncated at zero: c u N + ( µ u it , σ 2 ∼ c ) (2) it c d N + ( µ d it , σ 2 ∼ c ) it Following Gautier and Saout (2015) this also allows the means to depend on explanatory variables z it that modify the timing of the price adjustment: ′ µ u = z it λ u (3) it µ d ′ = it λ d z it

The model Estimation Stata xtss command Numerical example Conclusion References The model The final equation is for the frictionless price, which is observed when prices are amended. Letting x it denote a vector of exogenous explanatory variables: it β + u i + ǫ it , ǫ it ∼ iidN (0 , σ 2 ′ p ∗ it = x ǫ ) (4) where u i ∼ iidN (0 , σ 2 u ) is an individual specific random effect that contains unobserved product heterogeneity. In the context of price modeling, this process would arise as a log-linear expression under isoelastic demand and constant marginal costs where x it contains factor prices.

The model Estimation Stata xtss command Numerical example Conclusion References Estimation Letting I u it = 1 indicate a price rise and I d it = 1 a price fall, the observed price in (1) is: p it = p it − 1 + ( I u it + I d it )( p ∗ it − p it − 1 ) ′ Letting d it = x it β + u i − p it − 1 , from (4) the above becomes: ∆ p it = ( I u it + I d it )( d it + ǫ it ) (5) where the indicators � 1 d it + ǫ it > c u if I u it it = (6) d it + ǫ it ≤ c u 0 if it . � 1 d it + ǫ it < − c d if I d it it = (7) d it + ǫ it ≥ − c d 0 if it .

The model Estimation Stata xtss command Numerical example Conclusion References Estimation Analogous to a Tobit II model, OLS based on the amended prices will be inconsistent as E [ ǫ it | I u it = 1 ∪ I d it = 1] � = 0. Instead the model is estimated by maximum likelihood. Given the first-order Markovian property of the model, the the contribution of product i to the likelihood given p i 0 is: T � ∞ � L i = f (∆ p it | p it − 1 , u i , x it , z it ) f ( u i ) du i −∞ t =2 The integral in the above is approximated by Monte Carlo integration using Halton draws. To derive f (∆ p it | p it − 1 , u i , x it , z it ) we distinguish between the cases where prices rise, prices fall and prices remain constant.

The model Estimation Stata xtss command Numerical example Conclusion References Price rise Suppressing the dependence on u i , x it and z it to simply the exposition, the contribution to the likelihood of a price rise is: f (∆ p it , I u it = 1 | p it − 1 ) = Pr ( c u it < ∆ p it | ∆ p it ) f (∆ p it | p it − 1 ) From (2) and (5), the components in the above are: � ∆ p it − d it � 1 f (∆ p it | p it − 1 ) = φ σ ǫ σ ǫ Φ( ∆ p it − µ u ) − Φ( − µ u σ c ) it it Pr ( c u it < ∆ p it | ∆ p it ) = σ c Φ( µ u σ c ) it

The model Estimation Stata xtss command Numerical example Conclusion References Price fall The contribution to the likelihood of a price fall is: f (∆ p it , I d it = 1 | p it − 1 ) = Pr ( c d it < − ∆ p it | ∆ p it ) f (∆ p it | p it − 1 ) The first component in the above is: Φ( − ∆ p it − µ d ) − Φ( − µ d σ c ) it it Pr ( c d it < − ∆ p it | ∆ p it ) = σ c Φ( µ d σ c ) it

The model Estimation Stata xtss command Numerical example Conclusion References Price constancy The contribution from no change in price occurs when both it − p it − 1 < c u it − p it − 1 > − c d p ∗ it and p ∗ it ; from (6)-(7) this is: Pr ( I u it = 0 , I d it = 0 | p it − 1 ) = Pr ( ǫ it − c u < − d it , ǫ it + c d > − d it | p it − 1 ) it it � �� � � �� � ǫ 1 it ǫ 2 it As ǫ 1 it ≤ ǫ 2 it , the above simplifies to: = Pr ( ǫ 1 it < − d it | p it − 1 ) − Pr ( ǫ 2 it < − d it | p it − 1 ) (8) Evaluating the above requires the CDF of the sum of a normal and truncated normal random variable.

The model Estimation Stata xtss command Numerical example Conclusion References Price constancy The pdf’s of ǫ 1 and ǫ 2 can be derived using the convolution formula. Focusing on ǫ 1 , after considerable algebra this yields: � � � ǫ 1 + µ u � a + b ǫ 1 + µ u 1 f ( ǫ 1 ) = s Φ( µ u /σ c )Φ φ (9) s s The components in the above expression are: � = σ 2 ǫ + σ 2 s c s µ u a = (10) σ ǫ σ c − σ c b = σ ǫ

The model Estimation Stata xtss command Numerical example Conclusion References Price constancy Making the substitution z = ( ǫ 1 + µ u ) / s in (9) and integrating yields CDF: m + µ u � 1 s Pr ( ǫ 1 ≤ m ) = Φ ( a + bz ) φ ( z ) dz Φ( µ u /σ c ) −∞ Based on in Owen (1980), the above is: � � 1 + b 2 , m + µ u 1 a − b Pr ( ǫ 1 ≤ m ) = Φ( µ u /σ c )Φ 2 √ , ρ = √ s 1 + b 2 where Φ 2 ( x , y , ρ ) is the bivariate normal CDF.

The model Estimation Stata xtss command Numerical example Conclusion References Price constancy For ǫ 2 the result is based on µ d , the term z = ( ǫ 2 − µ d ) / s and correlation coefficient is negative. Substituting the expressions for s , a , b in (10), the probability of no change in price in (8) is given by: � � � � µ u µ u µ d σ c , − µ d it − d it it − d it √ √ √ √ Φ 2 σ c , c , Φ 2 c , − it σ c it σ c σ 2 ǫ + σ 2 σ 2 ǫ + σ 2 σ 2 ǫ + σ 2 σ 2 ǫ + σ 2 c c − Φ( µ u it /σ c ) Φ( µ d it /σ c ) This completes the derivation of f (∆ p it | p it − 1 , u i ). The resulting MLE will be consistent if N or T → ∞ .