GY460 Techniques of Spatial Analysis

1. GY460 Techniques of Spatial Analysis Lecture 5: Instrumental variablesNotes on the method to accompany student presentations of papers Steve Gibbons

2. Introduction and aims • Instrumental Variables/Two Stage Least Squares can provide consistent estimates when OLS is biased by omitted variables (or measurement error) • Techniques discussed in last lecture – differencing, fixed effects, discontinuity designs – are based on eliminating nuisance omitted variables by differencing the data • IV is based on finding source(s) of variation in the variable(s) of interest that is uncorrelated with the omitted variables • This lecture outlines the econometrics of IV • Best reading is Angrist and Pishke (2009) Chapter 4 • Seminar covers spatial applications

3. The demand/supply identification problem (1) • Demand and supply equations • If I regress q on p using equilibrium prices and quantities what do I estimate? • Equilibrium relationship between q and p. • This does not estimate the “structural” parameters of the demand or supply curve

4. The demand/supply identification problem (2) • But rearrange supply curve, and subsititute demand curve to obtain ‘reduced form’ • Now m changes prices via demand shifts, z changes prices via supply shifts • z can be used as an ‘instrument’ for p in the demand equation • m can be used as an ‘instrument’ for p in the supply equation

5. Basic IV model (1) • We are interested in the parameter beta • However, theoretical reasoning tells us • hence x is correlated with f, but we have no data to control for f. However, theory tells us z is uncorrelated with  • The reduced form is • Population regression of y on z is • Population regression of x on z is

6. Basic IV model (1) • Hence ratio of these regression coefficients is • IV estimator replaces these population parameters (Covariances) with sample analogues • You can also use the moment condition Cov(i,zi)= 0

7. Two Stage Least Squares/2SLS (1) • You can estimate in two steps. Consider the population models • 1st step: • 2nd step • 2SLS is the sample analogue of this, and generalises to the case of multiple instruments and additional exogenous variables

8. IV essentials • You have at least one instrument for each endogenous variable, and it is best to try and get a specification with only one endogenous variable. • Instruments must be uncorrelated with the error term in your main equation (conditional on other covariates) i.e. they can be theoretically be excluded from the second stage • You can’t test this if you only have as many instruments as regressors, but you can if you have more (Hansen-Sargan test) • Instruments are correlated with the endogenous variable! • You have to check that the instruments are jointly significant in the 1st stage regression: research should show the first stage and F-statistics • IV is consistent NOT unbiased. Bias in small samples may be large, and increases with weak instruments

9. IV with heterogeneous responses • IV is harder to interpret when the response of individuals (or other agents) is heterogeneous e.g. • If the response i is uncorrelated wth xi then OLS estimates the average of i in the population • But with IV, the predicted variation in x comes from a sub-group of the population who’s value of x is changed by the instrument (‘compliers’) • e.g. instrumenting “college attendance” with “living close to a college” estimates he affect for individuals who’s decision to participate is affected by travel costs • The causal effect for the compliers may be different from the causal effect of x in the population as a whole • In this context, IV gives rise to Local Average Treatment Effects (LATE – see Angrist and Pishke 2009 for details)

10. IV in the spatial context (1) • In the lecture on spatial dependence models we discussed the idea of using spatial ‘lags’ of x as an instrument for a spatial lag of y • Now we are thinking of the case where a spatial x variable is endogenous, probably because of spatial sorting i.e. • Z will make good instruments if uncorrelated with f(and u) and  0 • What things are likely to work, and which are likely to fail?

11. IV in the spatial context (2) • Evaluate these ideas: • Historical lags of city population as an instrument of current city population • Policies that have spatially differentiated impacts • Location of churches as an instrument for children’s attendance at a church school • Distance from a single east-west running border to predict trade penetration into the southern region from the northern region • Instrument college attendance with living close to a college • We will look at others in the seminar…