Identification and Estimation of Demand for Differentiated Products

Transcription

1 Identification and Estimation of Demand for Differentiated Products Jean-François Houde Cornell University & NBER September 29, 2017 Demand for Differentiated Products 1 / 52

2 Starting Point: The Characteristic Approach Ultimate goal: Measure the elasticity of substitution between goods. Why? Measure the degree of market power Predict the effect of proposed mergers Evaluate the value of new goods... Natural starting point: Linear inverse demand p jt = α 0 + k J α j,k q kt + ɛ jt Curse of dimensionality: Even with exogenous variation in q s, the number of parameters to estimate grows with the number of products. Solution: The characteristic approach (Lancaster 1966) Each product is defined as a bundle of characteristics: x j = {x 1,..., x k } Consumers assigned values to each characteristics, and choose the product maximize their utility given their budget. Demand for product j is function of the distribution of valuations for each attribute (finite dimension). Demand for Differentiated Products Introduction 2 / 52

3 Discrete Choice Models Two classes of models Local: Each product has only few close substitutes Global: Each product is equally substitutable with all others Global competition: Logit (McFadden 1974, Anderson et al. 1992) Indirect utility: u ij = δ j p j + ɛ ij }{{}}{{} average T1EV Demand (shares): Elasticity of substitution: exp((δ j p j )/σ ɛ ) s j = j J exp((δ j p j )/σ ɛ) η jk = ( ) 1 pk s j s k σ ɛ s j = 1 σ ɛ p k s j Two key properties: (i) Independence of Irrelevance Alternatives (IIA), and (ii) monopolistic competition in the limit (i.e. constant markup). Demand for Differentiated Products Introduction 3 / 52

4 Local competition: Vertical or horizontal differentiation? Definition: Horizontal differentiation: Consumers have different valuations of products available (even with equal prices) Vertical differentiation: Consumers would choose the same product if prices were equal. Horizontal Example: Linear city model (Hotelling 1929) Preferences for product j = 1, 2: u ij = δ p j λ l i x j where l i U(0, 1) is the location (or taste) of consumer i, and x j (0, 1) is the address of products (i.e. characteristic) and x 2 > x 1. Demand: s 1 = l 0 f (l i )dl i = λ(x 2 x 1 ) p 1 + p 2 2 where l is the marginal consumer (i.e. indifferent type). Demand for Differentiated Products Introduction 4 / 52

5 Local competition: Vertical or horizontal differentiation? Vertical Example: Quality ladder (Shaked and Sutton 1982, Bresnahan 1987) Preferences for product j = 1, 2: u ij = θ i x j p j where θ i U(0, λ) is the type of consumer i, and x j (0, 1) is the quality of products (i.e. characteristic) and x 2 > x 1. Demand: Type θ: θx1 p 1 = θx 2 p 2 s 1 = θ 0 f (θ i )dθ i = θ/λ = 1 p 2 p 1 λ x 2 x 1 Demand for Differentiated Products Introduction 5 / 52

6 Hybrid model: Random-coefficients The work-horse model is the random-utility model with heterogenous valuations for characteristics (aka random-coefficients): max j J x jβ i α i p j + ɛ ij Note 1: Despite the multiplicative form, this functional form nests some Hotelling-style models. Example: u ij = δ j p j λ(z j l i ) 2 = x j β i αp j. Note 2: The nested-logit model is a special case of the random-coefficient model: u ij = x j β + k 1(j G k )ν ik + ɛ ij where G k is a nest or product-segment (i.e. discrete attribute). Demand for Differentiated Products Introduction 6 / 52

7 From McFadden to BLP Data: Panel of market shares and product characteristics: {s t, p t, x t } t=1,...,t where t indexes a market, x t = {x jt,1,..., x jt,k } j=1,...,nt is a matrix of observed characteristics, and {p t, s t } = {p jt, s jt } j=1,...,nt is a matrix of endogenous prices and market shares. I will use w jt to denote a vector of price instruments (e.g. cost shifters, ownership, etc.) Market shares: s jt = q jt /M t where M t is the observed market size (exogenous). Examples: Cars: U.S. population Cereals: Population Avg. servings per month. Gasoline: Transportation needs (including car, bus, walk, etc.) Demand for Differentiated Products Aggregate demand for differentiated products 7 / 52

8 From McFadden to BLP Assumption: s jt is measured without error (very important). This often mean that you should aggregate data across time periods or markets to reduce the importance of measurement error in q jt. The data requirement in BLP is often underestimated... You don t need consumer-level data, but you need data on every consumers! Indirect utility function (Nevo 2001): u ijt = { K k=1 β i,kx jt,k α i p jt + ξ jt + ɛ ij If j 0 ɛ i0 Else. β i,k = β k + z i π k + η i,k α i = α + z i π p + η i,p z i F ( ) (known) and ν i N(0, Σ) (unknown) Average indirect utility: δ jt = x jt β αp jt + ξ jt. Demand for Differentiated Products Aggregate demand for differentiated products 8 / 52

9 Baseline Model: Exogenous Characteristics Consider first a model without prices and without demographics characteristics Demand: Linear random-coefficient with T1EV random utility shocks ( ) ( ) exp δ σ j δ t, x (2) jt + νi T x (2) jt t ; Σ = 1 + ( )df (ν J t j =1 exp δ j t + νi T x (2) i Σ) where δ jt = β 0 + x (1) jt β 1 + x (2) jt β 2 + ξ jt. The residual of the model is obtained from the inverse-demand function: ρ j (s t, x t ; θ) = σ 1 j ( s t, x (2) t ) ; Σ x jt β, j t where θ = (β, Σ). Demand for Differentiated Products The Identification Problem 9 / 52

10 Identifying Assumption Assumption: The unobserved attribute of each product is independent of the menu, x t, of characteristics available in market t, E[ξ jt x t ] = 0 (CMR). In practice, the model is estimated using a finite number (L) of unconditional moment restrictions, A j (x t ): E [ ρ j (s t, x t ; θ 0 ) A j (x t ) ] = 0 [( ( E s t, x (2) t ; Σ 0) x jt β σ 1 j ) ] A j (x t ) = 0. Gandhi & Houde (2016): How to construct relevant instruments to identify Σ? Stock & Wright (2000): Aj (x t ) is weak if the moment conditions are almost satisfied away from the true parameters. Demand for Differentiated Products The Identification Problem 10 / 52

11 What in the data identifies the model? Like any IV problem, we need to have enough instruments to estimate ( β, Σ). But what is a relevant/valid instrument? The CMR suggests that θ is identified by variation in the choice-set of consumers. Most identification sections describe the ideal experiment in which a new product enters, and steal market shares from existing products with similar attributes. This red-bus/blue-bus logic is correct, but we need to use it to construct IVs, otherwise the model will be weakly identified (even if the ideal experiment is present the data-set!) Demand for Differentiated Products The Identification Problem 11 / 52

12 Illustration of the Weak IV Problem Two detection tests: 1 Testing the wrong model: IIA hypothesis H 0 : E [ρ j (s t, x t β, Σ = 0) z jt ] = 0 ln s jt /s 0t = x jt β + γz jt + ξ jt H 0 : γ = 0 2 Local identification: Cragg-Donald rank test rank ( E [ ρ j (s t, x t ; θ) / θ T z jt ]) = m This test can be implemented in STATA (ivreg2 or ranktest). Monte-Carlo design: Sample: T = 100 and J = 15 Random utility with (independent) normal random-coefficients (K2 ) DGP: (xjt,k, ξ jt ) N(0, I ) Demand for Differentiated Products Illustration 12 / 52

13 Weak Identification in a Picture: IIA Test (A) IV: Sum of rivals characteristics (B) IV: Euclidean distance in x Residual qualities at Σ= Regression R 2 = Sum of rival characteristics Residual qualities at Σ= Regression R 2 = Euclidian distance (x) Takeaway: Independence of ξ jt and the distance of rival characteristics rules out the IIA hypothesis, but not the sum of rival characteristics. Demand for Differentiated Products Illustration 13 / 52

14 Distribution of ˆσ 2 with weak IVs Fraction Parameter estimates (exp) Shapiro-Wilk test for normality: (0). Width = 1. Demand for Differentiated Products Illustration 14 / 52

15 GMM Estimates with Weak IVs K 2 = 1 K 2 = 2 K 2 = 3 K 2 = 4 bias rmse bias rmse bias rmse bias rmse log σ log σ log σ log σ σ σ σ σ (Local-min) Range(J) Range(pv) Range(log σ) Rank-test p-value IIA-test p-value Demand for Differentiated Products Illustration 15 / 52

16 GMM Estimates with Weak IVs K 2 = 1 K 2 = 2 K 2 = 3 K 2 = 4 bias rmse bias rmse bias rmse bias rmse log σ log σ log σ log σ σ σ σ σ (Local-min) Range(J) Range(pv) Range(log σ) Rank-test p-value IIA-test p-value Demand for Differentiated Products Illustration 16 / 52

17 GMM Estimates with Weak IVs K 2 = 1 K 2 = 2 K 2 = 3 K 2 = 4 bias rmse bias rmse bias rmse bias rmse log σ log σ log σ log σ σ σ σ σ (Local-min) Range(J) Range(pv) Range(log σ) Rank-test p-value IIA-test p-value Demand for Differentiated Products Illustration 17 / 52

18 Non-Parametric Identification (Berry and Haile (2014)) To get a sense of how the model is identified, it is useful to consider an ideal setting. Infinitely many markets with exogenous changes in the choice-set facing consumers. Note: BH are interested in studying the identification of the demand system σ j (x t ); not the distribution f (ν i Σ). New normalization: ( σ 1 j s t, x (2)) = x (1) jt + ξ jt where x (1) jt is a scalar (i.e. special regressor). The BLP model written in this form correspond to a non-parametric instrumental variable model (Newey and Powell (2003)): x (1) jt = σ 1 j ( s t, x (2)) + ξ jt Demand for Differentiated Products Non-Parametric Identification 18 / 52

19 Non-Parametric Identification (Berry and Haile (2014)) Question 1: Is it possible to identify σ 1 j ( ) from the conditional mean restriction (i.e. E(ξ jt x t ) = 0)? To answer this we need to rely on the non-parametric analog of a rank condition. Do we have enough independent excluded instruments? Reduced-form: Applying the CMR to the inverse-demand equation, [ E[x (1) jt x t ] = E (s t, x (2)) ] x t + E[ξ jt x t ] x (1) jt σ 1 j [ = E σ 1 j (s t, x (2)) x t ] Completeness condition: For all functions B(s t ) with finite expectations, if E[B(s t ) x t ] = 0 almost surely, then B(s t ) = 0 a.s. If the demand function and distribution of x t satisfy this condition, then there is a unique σ 1 j ( ) that can be inverted from the reduced-form (see Theorem 1). Demand for Differentiated Products Non-Parametric Identification 19 / 52

20 Non-Parametric Identification (Berry and Haile (2014)) Question 2: Is there a unique demand function associated with σ 1 j (s t, x t )? The answer is yes in most mixed-logit models (see Berry (1994)) Fore more general demand systems, Berry, Gandhi, and Haile (2013) defined a new condition that is sufficient for existence and uniqueness of an inverse demand: connected substitutes Require that the index δjt weakly lower the market shares of all goods k j. See paper for more details... If both conditions are satisfied, the model is non-parametrically identified. The argument is only slightly more complicated with prices: The completeness condition (i.e. full rank) extends to the price IVs. Demand for Differentiated Products Non-Parametric Identification 20 / 52

21 Identification Back to the parametric model... The model is identified by the same logic: [ E σ 1 j E[ρ j (s t, x t ; θ) x t ] = 0, iff θ = θ 0 ( s t, x (2) t ; Σ 0) ] x t } {{ } Reduced-form β 0 x (1) jt β 1 x (2) jt β 2 = 0 Takeaway: The presence of a special regressor x (1) jt implies that x (1) can be used as excluded instruments for the endogenous shares. j,t Demand for Differentiated Products Parametric Identification 21 / 52

22 How to select the instruments? Since dim(x t ) >> dim(σ) = m, any transformation of x t = {x 1t,..., x Jt,t} can be used to construct valid moments: E[ρ j (s t, x t ; θ) A L j (x t )] = 0, where dim(a L j (x t )) = L m Donald, Imbens, and Newey (2003): Conditional moment restriction is equivalent to a countable number of unconditional moment restrictions (aka IVs), E[ρ j (s t, x t ; θ) A L j (x t )] = 0 Iff E[ρ j (s t, x t ; θ) x t ] = 0, where the instruments A L j (x t) correspond to basis functions spanning the space of x t (dimension L). In our context, a necessary condition for this equivalence is that the reduced-form of the model can be approximated by A L j (x t) as L : [ E {g j (x t ) A L j (x t )γ L } 2] 0 Demand for Differentiated Products Parametric Identification 22 / 52

23 Curse of Dimensionality Problem Curse of Dimensionality: The reduced-form is a product-specific function of the entire menu of product characteristics. As J, both the number of arguments and the number of functions to approximate increase. This is a different point than the one raised by Armstrong (2015), which is about the weakness of the price instruments as J. Without further restrictions, we cannot directly use the insights of BH to construct relevant IVs Our approach: Reduce the dimensionality of the problem by exploiting the symmetry of the demand function (implied by the linearity of the random utility model) Demand for Differentiated Products Parametric Identification 23 / 52

24 What does the characteristic structure imply for the reduced-form of the model? Market-structure facing product j (dropping t): (( ) ( )) (w j, w j ) δ j, x (2) j, δ j, x (2) j Properties of the linear-in-characteristics model: Symmetry: σ j (w j, w j) = σ k (w j, w j) k j Anonymity: σ (w j, w j ) = σ ( w j, w ρ( j) ) ρ Translation invariant: for any c R K σ (w j + (0, c), w j + (0, c)) = σ (w j, w j ) Demand for Differentiated Products Parametric Identification 24 / 52

25 Re-Express the Demand System Express the state of the market in differences relative to j and treat the outside option just like any other product. Characteristic differences: New normalization: Product k attributes: ωj,k = d (2) j,k = x (2) k x (2) j exp(δ j ) τ j = 1 + j exp(δ j = 0,..., n. j ), ( ) τ k, d (2) jt,k Demand for product j is a fully exchangeable function of ω j : σ (w j, w j ) = D(ω j ) where ω j = {ω j,0,..., ω j,j 1, ω j,j+1,..., ω j,n }. Demand for Differentiated Products Parametric Identification 25 / 52

26 Main Theory Result Define the exogenous state of the market facing product j: d j,k = x k x j d j = (d j,0,..., d j,j 1, d j,j+1,..., d j,n ) Theorem If the distribution of {ξ j } j=1,...,n is exchangeable (conditional on x jt ), then the reduced form becomes [ ( E s, x (2) ; Σ 0) ] x = g (d j ) σ 1 j where g is a symmetric function of the state vector. Implication: g is a vector symmetric function (see Briand 2009) Demand for Differentiated Products Parametric Identification 26 / 52

27 Why is it useful? 1 Curse of dimensionality: The number of basis functions necessary to approximate the reduced-form is independent of the number of products (Pakes (1994), Altonji and Matzkin (2005)). 2 Example: Single dimension d jt = {x 1t x jt, x 2t x jt,..., x Jt,t x jt } First-order approximation of g(d): g(d jt ) γj 1 d jt,j = γ1 d jt,j j j Second-order approximation of g(d): g(d jt ) γj 1 d jt,j + γj 2 (d jt,j )2 + γ 3 d jt,j j j j 2 = γ 1 d jt,j + γ 2 j j (d jt,j ) 2 + γ 3 d jt,j j 2 Demand for Differentiated Products Parametric Identification 27 / 52

28 Closing the loop: What is a relevant IV? Let A j (x t ) be an L vector of basis functions summarizing the empirical distribution of characteristic differences: {d jt,k } k=0,...,jt. Differentiation IV: These functions are moments describing the relative isolation of each product in characteristic space. Donald, Imbens, and Newey (2003): Using basis functions directly as IVs, is asymptotically equivalent to approximating the optimal IV. Recommended practice is to use low-order basis functions (Donald, Imbens, and Newey 2008). Demand for Differentiated Products Differentiation IVs 28 / 52

29 Practical Suggestions: Polynomial Basis Note: In general, the first-order basis is weak because it does not vary across products within markets (i.e. sum). In practice you might still want to include the first-order basis terms when you have a lot of entry/exit of products (perhaps interacting with other distance measures). Could be useful also to identify random coefficient on the intercept (rough intuition). Single dimension measures of differentiation Quadratic: A j (x t ) = ( ) 2 djt,j k j Note: z jt,k is the Euclidian distance between product j and its rivals in market t along dimension k. Adding interaction terms: Covariance: A j (x t ) = j d k jt,j d l jt,j Demand for Differentiated Products Differentiation IVs 29 / 52

30 Practical Suggestions: Histogram Basis Note: This approach is advisable only in very large samples, and when the goal is to estimate a very flexible distribution of RCs. Single dimension measure of differentiation = Number of rivals in discrete bins ( ) A j (x t ) = 1 djt,j k < κ l j l=1,...,l Multi-dimension measure of differentiation: ( ) ( ) A j (x t ) = djt,j k < κ l 1 djt,j k < κ l j 1 l=1,...,l,l =1,...,L Demand for Differentiated Products Differentiation IVs 30 / 52

31 Practical Suggestions: Local Basis Note: In most parametric models, the inverse demand is function of characteristics of close-by rivals. Therefore, in the previous histogram basis, we should be focussing on local rivals. Single dimension measure of differentiation = Number of nearby rivals along each dimension A j (x t ) = ( ) 1 djt,j k < κ k, e.g. κ k = sd(x jt,k ) j Multi-dimension measure of differentiation: A j (x t ) = ( ) 1 djt,j k < κ k d jt,l, e.g. κ k = sd(x jt,k ) j When x jt,k is discrete, this basis function boils down to the familiar Nested-logit IVs. Number of competitors and characteristics of rivals within segment. See Bresnahan, Stern, and Trajtenberg (1997). Demand for Differentiated Products Differentiation IVs 31 / 52

32 Practical Suggestions: Demographics In many settings, product characteristics are fixed across markets, but the distribution of consumer types vary (e.g. Nevo 2001). To fix ideas, focus on a single non-linear characteristics x (2) j Consumer valuation for x (2) j is β it = z it π + ν i where ν i N(0, σx). 2 Assumption: The distribution of demographics across markets is known, and can be decomposed as follows: z it = µ t + sd t e it, where e it F ( ) and F ( ) is common across markets. Example: BLP95 assume that the income distribution is log-normal with market-specific mean/variance. Demand for Differentiated Products Differentiation IVs 32 / 52

33 Practical Suggestions: Demographics Demand function: σ jt (δ t, x (2) π, σ x ) = ( ) exp δ jt + z it πx (2) j + ν i x (2) j = 1 + ( )df t (z it )φ(ν i ; Σ) j exp δ j t + z it πx (2) j + ν ix (2) j ( ) exp δ jt + πe it σ t x (2) j + πµ t x (2) + ν i x (2) j = 1 + j exp ( δ j t + πe it σ t x (2) j + πµ tx (2) j + ν ix (2) j )df (e it )φ(ν i ; Σ) = σ j (δ j, x (2), σ t x (2), µ t x (2) π, σ }{{} x ) new state variables Implication: Demand is a symmetric function of characteristic differences and demographic moments: D(ω t, d (2), σ t d (2), µ t d (2) θ). The previous result therefore applies to the reduced-form of this transformed model: [ ] E σ 1 jt (s t, x (2) π, σ x ) x t, µ t, σ t = g(d t, µ t d (2), σ t d (2) ) Demand for Differentiated Products Differentiation IVs 33 / 52

34 Practical Suggestions: Demographics Differentiation IVs with demographics: A t (x t, µ t, σ t ) = ( ) 1 djt,j k < κ k µ t j A t (x t, µ t, σ t ) = ( ) 1 djt,j k < κ k σ t j A t (x t, µ t, σ t ) = ( ) 1 djt,j k < κ k σ t djt,j l j When the distribution of demographics can be standardized across markets, this characterization is exact. More generally, Differentiation IVs should be interacted with rich moments of the distribution of consumer characteristics. Example: Miravete, Seim, and Thurk (2017) Combine nested-logit type instruments, with moments of the distribution of demographics across stores. Demand for Differentiated Products Differentiation IVs 34 / 52

35 Experiment 1: Independent Random Coefficients Random coefficient model: u ijt = δ jt + K k=1 v ik x (2) jt,k + ε ijt, v i N(0, σ 2 xi ). Data: Panel structure: 100 markets 15 products Characteristics: (ξ jt, x jt ) N(0, I). Dimension: x jt = K + 1 Monte-Carlo replications = 1,000 Differentiation IVs (K + 1): Quadratic: Aj (x t ) = J t j =1 ( d k jt,j ) 2, k = 1,..., K Demand for Differentiated Products Monte-Carlo Simulations 35 / 52

36 Simulation Results: Quadratic Differentiation IVs K 2 = 1 K 2 = 2 K 3 = 3 K 3 = 4 bias rmse bias rmse bias rmse bias rmse log σ log σ log σ log σ σ σ σ σ (Local) Rank-test pv IIA-test pv Demand for Differentiated Products Monte-Carlo Simulations 36 / 52

37 Experiment 2: Correlated Random Coefficients Consumer heterogeneity: β (2) i N(β (2), Σ) 4 dimensions 10 non-linear parameters (choleski) Panel structure: 100 markets 50 products Differentiation IVs: Second-order polynomials (with interactions): for all characteristics k <= l. J t ) A j (x t ) = (d jt,j k d jt,j l j =1 Demand for Differentiated Products Monte-Carlo Simulations 37 / 52

38 Simulation Results: Correlated Random-Coefficients Σ,1 Σ,2 Σ,3 Σ,4 Σ 1, Σ 2, Σ 3, Σ 4, Σ 1, Σ 2, Σ 3, Σ 4, IIA test (F) Cragg-Donald stat Nb endo Nb IVs Demand for Differentiated Products Monte-Carlo Simulations 38 / 52

39 How to account for endogenous prices? Quality-ladder example: u ijt = δ jt α i p jt + ε ijt where α i = σ p y 1 i, and log(y i ) N(µ y, σ y ) (known). Excluded price instruments: w t = {w jt } j=1,...,jt Reduced-form: [ E σ 1 j ( ] st, x t, p t σp) 0 x t, w t g(d x j, d p j ), where the inequality is due to the simultaneity of prices and ξ jt (BLP, 1995). Demand for Differentiated Products Endogenous prices 39 / 52

40 How to incorporate endogenous prices? Heuristic solution: Distribute the expectation for price inside of the inverse-demand (BLP, 1999): [ ] [ ] E σ 1 j (s t, p t, x (2) t ; Σ) x t, w t E σ 1 j (s t, ˆp t, x (2) t ; Σ) x t, ˆp t where ˆd p jt,k = E(p kt w kt ) E(p jt w jt ). = g(d x jt, d ˆp jt ) Demand for Differentiated Products Endogenous prices 40 / 52

41 Experiment 3: Differentiation IVs with Endogenous Prices Example with cost shifter 1 Exogenous price index (OLS): 2 Differentiation IV: Quadratic j ˆp jt = ˆπ 0 + ˆπ 1 x jt + ˆπ 2 ω jt ) 2 ( ) 2 (d ˆp jt,j and d ˆp jt,j d jt,j j where d jt,j = (d x jt,j, d ˆp jt,j ). 3 Differentiation IV: Local ( ) d ˆp jt,j < sd(ˆp jt ) j and ) ( d ˆp jt,j < sd(ˆp jt ) d jt,j j Demand for Differentiated Products Endogenous prices 41 / 52

42 Distribution of ˆσ p with weak and strong IVs Fraction Kernel density Random coefficient parameter (Price) Diff IV: Market Diff IV: Local Diff IV: Quadratic Dash vertical line = True parameter value Demand for Differentiated Products Endogenous prices 42 / 52

43 GMM estimates with endogenous prices True Diff. IV = Local Diff. IV = Quadratic Diff. IV = Sum bias se rmse bias se rmse bias se rmse σ p β β x β p Demand for Differentiated Products Endogenous prices 43 / 52

44 GMM estimates with endogenous prices Diff. IV = Local Quadratic Diff. IV = Sum Frequency conv IIA-test p-value st-stage F-test: Price st-stage F-test: Jacobian Cond. 1st-stage F-test: Price Cond. 1st-stage F-test: Jacobian Cragg-Donald statistics Stock-Yogo size CV (10%) Nb. endogenous variables Nb. IVs Note: The Conditional 1st-stage F-test statistic is the Weak IV test proposed by Angrist and Pischke for multiple endogenous variables. Demand for Differentiated Products Endogenous prices 44 / 52

45 Experiment 4: Natural Experiments Hotelling example: Exogenous entry of a new product (x = 5) u ijmt = δ jmt λ ν i x jmt + ɛ ijmt Three-way panel: product j, market m, and time (t = 0, 1). Treatment variable: D jm = 1 ( x jm 5 < Cutoff) Reduced-form: Difference-in-difference regression σ 1 j (s t, x t, p t θ 0 ) = µ jm + τ t + γd jm 1(t = 1) + ξ jmt GMM: DiD IVs Linear characteristics: x (1) jmt = Market/Product FE + After Dummy Differentiation IV: zjmt = D jm 1(t = 1) ˆθgmm is identified from the DiD variation in z jmt. Demand for Differentiated Products Natural Experiments 45 / 52

46 Natural Experiment: Hotelling Example DGP: δ jmt = ξ jm + ξ jmt, where E( ξ jm x m ) 0 Diff-in-Diff specification: z jmt = {Product Dummy jm, 1(t = 1), 1( x jm 5 < 1)1(t = 1)} kdensity theta x Diff. IV + FE Diff. IV BLP (1999) Demand for Differentiated Products Natural Experiments 46 / 52

47 Review of Nonlinear GMM The GMM problem is defined as: min = β,σ ρ(θ) ZŴ 1 Z ρ(θ) s.t. ρ j (s t, x t θ) = σ 1 j (s t, x (2) t Σ) x jt β The FOC of the problem is (or moment conditions): where ρ(θ) θ function. In our context: ρ(θ) ZŴ 1 Z ρ(θ) = 0 θ is a n m matrix containing the Jacobian of the residual ρ j (s t, x t θ) θ = { xjt,k If θ k = β k σ 1 j (s t,x tθ k ) Σ k Else. The derivative of the inverse demand can be computed using the implicit function theorem (see Nevo (2001)). Demand for Differentiated Products Inference and testing for weak IVs 47 / 52

48 Review of Nonlinear GMM Let J jt (θ) = ρ j (s t,x t θ) θ denotes the matrix of Jacobian. Notice that the moment conditions imposed by GMM correspond to the moment conditions associated with a linear approximation of the model. The moment conditions at θ gmm is: J(θ) ZŴ 1 Z ρ(θ) = 0 This is the moment condition of the following linear IV regression: where b = 0. ρ jt (θ) = J jt (θ)b + Error = k (θ k θ 0 k ) ρ j(s t, x t θ) θ + Error Therefore, b gmm = E(θ gmm θ 0 ) = 0 (i.e. GMM is consistent). Demand for Differentiated Products Inference and testing for weak IVs 48 / 52

49 Gauss-Newon Regression The linear representation can be used to construct a Gauss-Newton algorithm to estimate {ˆθ}. Iteration k 1: ρ jt (θ) = J jt (θ)b + Error 1 Invert demand: σ 1 jt (s t, x t θ k 1 ) and J jt (s t, x t θ k 1 ) 2 Estimate {ˆb k } by linear GMM. 3 If ˆb k < ε stop. Else, update θ k = θ k 1 + ˆb k, and repeat steps (1)-(3). With strong instruments, this procedure typically requires less than 5 iterations. Weak IVs lead to severe numerical problems (e.g. Knittle and Metaxoglou (2014), Dube et al. (2012)) Why? The central-limit theorem does no hold, and the quadratic approximation is bad. Demand for Differentiated Products Inference and testing for weak IVs 49 / 52

50 Using the Gauss-Newton Regression for Inference The Gauss-Newton regression is also useful to conduct inference Let ˆθ and Ŵ denote GMM estimate and weighting matrix (estimated using Julia or any other non-linear optimization package) Since non-linear GMM is equivalent to linear IV at the solution, we can conduct inference on ˆθ using the GNR: ρ jt (ˆθ) = J jt (ˆθ)ˆb + Error Note: ˆb must be zero if ˆθ is the GMM solution (i.e. you must be using the same weighing matrix in Julia or Matlab as in STATA or R to run this regression) But, variance-covariance matrix of ˆθ is the same as the variance-covariance matrix of ˆθ. Therefore, you can use standard statistical routines to calculate standard errors and conduct hypothesis tests on θ (e.g. cluster standard errors, equality of parameters, etc.). Demand for Differentiated Products Inference and testing for weak IVs 50 / 52

51 Using the Gauss-Newton Regression for to evaluate the Strength of IVs This insight is particularly useful to evaluate the relevance of the instruments Some weak identification tests are non trivial to code, and are standard in STATA and R Examples: IVREG2 in STATA reports the Cragg-Donald and the Sanderson-Windmeijer (SW) first-stage tests. Recommeded procedure: Use Julia s non-linear optimization packages to solve the GMM problem Compute the Jacobian of the inverse demand Expor the Jacobian and weighting matrix to STATA or R (or even better load R in Julia...) Perform inference and weak IV tests using the Gauss-Newton regression Demand for Differentiated Products Inference and testing for weak IVs 51 / 52

52 Ex-Ante Weak IV test: IIA Hypothesis Warning: The weak IV test based on the Jacobian function at ˆθ are not consistent when the instruments are too weak. You should therefore not put too much weight on the p-values, and combine your analysis with the IIA test. A strong instrument for Σ is able to reject the wrong model (Stock and Wright, 2000) Under H 0 : Σ = 0, the inverse demand equation is independent of x j : σ 1 jt (s t, x t ; Σ = 0) = ln s jt /s 0t = x jt β + z jt γ + ξ jt Standard test statistics for H 0 : γ = 0, can be used to test null hypothesis of IIA preferences With endogenous prices, this test is equivalent to the J-test at Σ = 0. In practice, you should report both this ex-ante weak IV test, and the ex-post specification tests. Demand for Differentiated Products Inference and testing for weak IVs 52 / 52

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