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Simultaneity Issues in Ordinary Least Squares Amine Ouazad Ass. Prof. of Economics Recap from last sessions • OLS is consistent under the linearity assumptions, the full rank assumption, and the exogeneity of the covariates assumption. • The exogeneity of the covariates (A3) is violated whenever: 1. 2. 3. • • • There is an omitted variable in the residual which is correlated with the covariates. (last session) There is measurement error. (last session) There is a reverse causality or simultaneity problem (this session). These three issues cause identification problems: even if sample size is infinite, the estimator does not come arbitrarily close to the true value. The OLS estimator is inconsistent. The OLS estimator is biased. Outline 1. Supply/demand estimation 2. Simultaneous Equation Model (X Rated) 3. Bruce Sacerdote, Peer effects with random assignment: Results for Dartmouth roommates, Quarterly Journal of Economics, 2001. SUPPLY AND DEMAND ESTIMATION 100 80 60 40 0 20 1940 1960 1980 Year 2000 2020 1940 1960 Current US $ 1980 Year 2000 2020 Total production in Barrels per day 30000 0 40 60 80 40000 50000 60000 70000 Total production in Barrels per day 20 80000 100 100 80 60 40 0 20 30000 40000 50000 60000 70000 Total production in Barrels per day Source: BP Statistical Review, 2009. Forecastchart.com 80000 . eststo: regress totalquantity price . esttab, se r2 (1) totalquant~y price 481.3*** (78.93) _cons 52098.0*** (2224.4) N R-sq 44 0.470 Standard errors in parentheses * p<0.05, ** p<0.01, *** p<0.001 Supply/demand estimation The problem • pt :price at time t. • qt : quantity at time t. • Specification: qt = a + b pt + e. • The OLS regression of qt on pt does not satisfy A3 because there is a correlation between price changes and the unobservables e. Full model = + + = − + • qt, pt: endogenous variables. • e: supply shock, u: demand shock. Market equilibrium: Notice the effect of supply and demand shocks on price and quantity. − − = + + + + = − ( − ) + + Hence Cov(pt,et) is non zero and the regression of quantity on prices does not yield a consistent estimator of either demand or supply. Exogenous/Endogenous • Greek: ενδογενής, meaning "proceeding from within" ("ενδο"=inside "-γενής"=coming from), the complement of exogenous (Greek: εξωγενής exo, "έξω"= outside) "proceeding from outside". • Definition of exogenous/endogenous depends on the model. For instance, in the previous model, if Structural, Identifiable Parameters • Structural parameters: (a,b,c,d,Var(e),Var(u)) • Identifiable parameters: − + − , , , Var(− − ) + + + + – These parameters are the mean of pt, the mean of qt, the variance of pt, the variance of qt. • Hence there are 4 identifiable parameters, and 6 structural parameters…. The model is not identified. Observational equivalence, example = 1 + 2 + = 4 − 5 + • With Cov(et,ut) = 0, Var(et) = 1, Var(ut) =1. • Then, the following demand and supply schedule gives the same distributions of prices, quantities, and correlation between price and quantity: = 1 + 2 + 1 2 + = 2 − + 3 3 • How did I find this?? Solving the simultaneity problem Intuition • If you have a variable that affects demand without affecting supply, then it is possible to identify the supply curve. • If you have a variable that affects supply without affecting demand, then it is possible to identify the demand curve. • Here we have this: – temperature affects only supply ! – We are able to estimate the demand curve. How ? Consider the following: = ( , ) ( , ) • Prove that this covariance is equal to b. – Under what assumption? • z is called a supply shifter. – A supply shifter identifies demand. • Question: what if you had one variable that affects demand without affecting supply? Stata application ivreg quantity (price = temperature) • This regression estimates the demand curve, since temperature affects only supply. • This is called an instrumental variable regression, to be seen later in econometrics A. (Reference: William Greene, Simultaneous Equations Model) SIMULTANEOUS EQUATIONS Structural form of the model • • • • yt1,ytM are the endogenous variables xt1, xtK are the exogenous variables. et1,…,etM are the structural residuals/shocks/unobservables. t: time periods. In matrix form: ′ Γ + ′ = ′ Reduced form of the model ′ = −′ Γ −1 + ′Γ −1 • Joy! This reduced form model can be estimated as is, with M separate equations, the OLS estimator of the regression of each element of yt on the xt is consistent. • But wait a minute: from Γ −1 it is not possible to recover all the elements of B and Γ . Exercise • Write the structural form of the model for the oil example. • Hints: – There are 2 equations. – 2 Endogenous variables: pt qt. – 1 Exogenous variable: the constant. • Write the reduced form model using the previous formula. Do we find the same solution? Matrix form notation: Structural model Γ + = • Y : TxM matrix. T rows, M columns. – M = 2 in the oil example. • X: TxK matrix. – K = 1 in the oil example. • E : TxM vector. • Exogeneity E(E|X) = 0 and E(E’E|X) = S. Matrix form notation: Reduced form model = Π + • P : the matrix of reduced form parameters. (KxM matrix). • V : the vector of residuals, with variancecovariance matrix W. The var-cov matrix has size M. Identification of Reduced Form parameters • Parameters in the structural form model: – M*M + K*M + ½ M(M+1) – G matrix, B matrix, S matrix. • Parameters in the reduced form model: – K*M + ½ M(M+1) – P matrix, W matrix. • Aie ! M*M parameters ‘too many’ ! Solutions? 1. Normalizations: – make the coefficient of each independent variable equal to 1. The number of excess parameters is then M(M-1). 2. Identities & Restrictions: – Pin down relationships between parameters. 3. Exclusions: – Political events have an effect on supply, but not on world demand. 4. Restrictions on the variance covariance matrix: – Assume 0 correlation between disturbances in the reduced form model. Notations for equation j • Considering equation j in isolation… • We set the coefficient on yj equal to 1. • We are going to exclude endogenous variables (exactly Mj* variables) and exclude exogenous variables (Kj* variables). Equation j Finding the structural parameters Pj*gj=pj* • This equation gives the structural parameters of equation j. It has Kj* equations (the rows of Pj*) and Mj unknowns (the coefficients of the structural parameters). Order condition Kj* greater or equal than Mj • The number of exogenous variables excluded in equation j must be at least as large as the number of endogenous variables included in equation j. • Relationship with agricultural product exercise?? Rank condition rank[Pj*] = Mj • This condition imposes a restriction on the submatrix of the reduced-form coefficient matrix. Deducing the coefficients of the exogenous variables bj=pj-Π gj • This equation gives the coefficients of the exogenous covariates of equation j as a function of known quantities. Take Away • With simultaneous equations, only the reduced-form model is typically identified in OLS. – You cannot interpret the results of an OLS of the structural model (where endogenous variables are in the covariates). • However, by making suitable assumptions on the exclusion of exogenous variables, you can identify the model. Take Away • 4 steps: 1. Estimate the reduced form model. 2. Make the necessary exclusion restrictions. 3. Write the structural parameters as a function of the reduced form parameters. 4. Solve the system of equations. SOCIAL INTERACTIONS: SACERDOTE QJE 2001 The question • What is the impact of a roommate’s characteristic and behavior on your GPA/behavior ? – Characteristic: SAT score (before university), gender, age, any x that is not changing over time. – Behavior: any variable that is a choice, such as choice of major, achievement. The issues • Regressing my GPA on the GPA of my roommate has a number of problems… – First, the GPA of my roommate is also determined by my GPA: simultaneity bias. (A3) – Second, if roommates are not randomly allocated, and if, for instance, having a male roommate is correlated with having a drinking roommate, then: omitted variable bias. (A3) – Third, there are some common shocks that affect both me and my roommate at the same time: correlation of the error terms. (A4) – Fourth, the effect of my roommate may depend on my characteristics, and also on his other characteristics: non linearity. (A1) The structural model • Academic ability, Measurement error, Grade point average of the roommate, residual. • Assumptions A1,A2 are maintained. The exogeneity of the GPA of the other roommate (A3) is violated. The reduced form model • Notice that right-hand side variables are purely exogenous. • Identification problem is given pi_0, pi_1, and pi_2, can we “get” the structural parameters? Without any more constraint, no. • Notice that there will be correlation of the residuals across individuals. Identification problem • In the reduced form, assuming A1,A2,A3, the OLS is consistent. A4 is violated (more on this in the next session). – regress gpa gpa_roommate characteristics characteristics_roommate gives consistent and unbiased estimates of the effects. • (To correct for A4, add option cluster(room) if room is a variable indicating the room number). Randomization • Individuals are randomized conditionally on their stated preferences. Violation of A3? Not if the preferences are present as covariates. – Conditional randomization, E(e|X) = 0. • Method of randomization? Results Checking Linearity (A1) Take Away from this session • Spot Reverse Causality issues in papers! – Sometimes mild, sometimes very severe (demand/supply, social interactions) • You can solve the problem by instrumenting the endogenous variable with a variable that affects the variable without affecting the outcome (a demand shifter for supply, a supply shifter for demand). OLS: Three identification issues 1. Measurement error 2. Omitted variable bias 3. Reverse causality/Simultaneity