Regression Analysis Tutorial 228 LECTURE / DISCUSSION. Simultaneous Equations

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1 Regression Analysis Tutorial 228 LECTURE / DISCUSSION Simultaneous Equations

2 Regression Analysis Tutorial 229 Simultaneous Equations Example 1 Gas consumption = " + $(thermostat setting) + 2(sq ft) + g Thermostat setting = 0 + N(income) + µ g and µ are correlated Gas consumption OLS estimate true Thermostat setting Problem: Because µ is correlated with g, and µ affects thermostat setting, thermostat setting is correlated with g

3 Regression Analysis Tutorial 230 Example 2 Price elasticity of telecommunications demand: MOU ' % (price) % (Nemply) % g Price ' % (MOU) % (region dum) % µ g and µ are correlated MOU OLS true Price Demand Problem: Because µ is correlated with g, and µ affects price, price is correlated with g

4 Regression Analysis Tutorial 231 Example 3 Aggregate demand and supply: Demand: Q = " + $P + g Supply: P = 8 + 2Q + µ

5 Regression Analysis Tutorial 232 Solution: Apply IV/2SLS M ' % P % N % g P ' % M % R % µ Endogenous: M, P Exogenous: R, N Step 1: Regress endogenous variables against all the exogenous variables M = a + br + cn + e P = d + fr + gn + u Get ˆM,ˆP Step 2: Regress original equations, replacing endogenous explanatory variables M and P with predicted values ˆM and ˆP M ' % ˆP % N % g ( P ' % ˆM % R % µ (

6 Regression Analysis Tutorial 233 Or apply IV directly Regress M = " + $P + 2N + g with instruments N and R and P = 8 + NM + 0R + µ with instruments N and R TSP commands: or 2sls(inst = (c,n,r)) m c,p,n ; 2sls(inst = (c,n,r)) p c,m,r ; inst m c,p,n invr c,n,r ; inst p c,m,r invr c,n,r ;

7 Regression Analysis Tutorial 234 Reduced-Form Equation Structural model: causal relationships M = " + $P + 2N + g P = 8 + NM + 0R + µ Structural parameters: ", $, 2, 8, N, 0 Reduced-form equations: endogenous variables expressed as a function of exogenous variables M ' M ' % ( % M % R % µ) % N % g M ' % % M % R % N % µ % g ( )M ' % % R % N % µ % g % % R % N % M = a + b R + c N + e µ % g

8 Regression Analysis Tutorial 235 Similarly, P ' % % R % N % g % µ P = d + f R + g N + u Reduced form parameters: a, b, c, d, f, g

9 Regression Analysis Tutorial Reduced-form equations are estimated in first step of 2SLS Structural equations are estimated in second step of 2SLS 2 Reduced-form equations can be estimated by OLS because exogenous variables are uncorrelated with errors 3 There is a relation between the reduced-form parameters and the structural parameters For example: c '

10 Regression Analysis Tutorial 237 Reduced form parameters give the full effect of a change in an exogenous variable N rises by 1 unit: M = " + $P + 2N + g P = 8 + NM + 0R + µ M increases by 2 P increases by N2 M increases by $[N2] P increases by N[$N2] = $N 2 2 M increases by $[$N 2 2] = $ 2 N 2 2 P increases by N[$ 2 N 2 2] = $ 2 N 3 2 M increases by $[$ 2 N 3 2] = $ 3 N 3 2 Total effect on M % % ( ) 2 % ( ) 3 % ' j 4 t'0 ( ) t '

11 Regression Analysis Tutorial 238 Total effect on P % 2 % 2 3 ' % ( ) % ( ) 2 ' j 4 t'0 ( ) t '

12 Regression Analysis Tutorial 239 Indirect Least Squares Sometimes, structural parameters can be calculated from reduced-form parameters a ' % d ' % b ' f ' c ' g ' Six equations, six unknowns b f ' g c ' etc

13 Regression Analysis Tutorial 240 Example: if â ' 4 ˆd ' 8 ˆb ' 3 ˆf ' 2 ĉ ' 7 ĝ ' 35 then ˆ '&8 ˆ ' 6 ˆ ' 15 ˆ ' 050 ˆ ' 175 ˆ ' 050 Note: ILS gives same estimates as 2SLS

14 Regression Analysis Tutorial 241 Cannot Always Apply ILS I Add an exogenous variable: M ' % P % N % I % g P ' % M % R % µ 7 structural parameters Reduced-form equations: M = a + bn + ci + dr + e P = f + gn + hi + RR + u 8 reduced-form parameters Eight equations for seven unknowns No solution Can still apply 2SLS 2SLS finds structural parameters that best fit the reduced form parameters

15 Regression Analysis Tutorial 242 II Omit an exogenous variable: M ' % P % g P ' % M % R % µ 5 structural parameters Reduced-form equations M = a + br + e P = c + dr + u 4 reduced-form parameters Four equations, five unknowns Infinite number of solutions

16 Regression Analysis Tutorial 243 Also: Step 1: Cannot apply 2SLS in Case II ˆM ' â % ˆbR ˆP ' ĉ % ˆdR Step 2: M ' % ˆP % g P ' % ˆM % R % µ ' % â % ˆbR % R % µ ' ( % â) % ( ˆb % )R % µ Cannot estimate N and 0 separately

17 Regression Analysis Tutorial 244 Identification Name Estimation method Just identified NRFP = NSP 2SLS = ILS Over identified NRFP > NSP 2SLS Under identified NRFP < NSP Cannot estimate NRFP: Number of reduced form parameters NSP: Number of structural parameters

18 Regression Analysis Tutorial 245 Efficiency 2SLS gives unbiased estimate But 2SLS ignores information contained in correlation between errors So, 2SLS is not efficient M ' % P % N % g P ' % M % R % µ g, µ correlated If we knew µ was high, we would know that g is probably also high and hence M is higher than predicted from P and N only Similarly for g and P

19 Regression Analysis Tutorial 246 3SLS: An Efficient Estimator Step 1: Estimate reduced form equations Get ˆM, ˆP Step 2: Estimate structural equations with ˆM and ˆP Save residuals from these regressions, labeled ĝ and ˆµ Step 3: Re-estimate structural equations with ĝ and ˆµ included as explanatory variables M ' % ˆP % N % ˆµ % g (( P ' % ˆM % R % ĝ % µ (( Because g and µ are correlated, ˆµ provides information for explaining M and ĝ provides information for explaining P Including this information makes the estimates better

20 Regression Analysis Tutorial 247 Special Case: Seemingly Unrelated Equations Y n ' % X n % g n W n ' % S n % µ n No endogenous explanatory variables But: g n and µ n are correlated Example: Y n = temperature in San Francisco W n = temperature in Monterey 3SLS can be done in two steps, because reduced-form equations are the same as the structural equations Step 1: Estimate equations by OLS Get residuals ĝ n and ˆµ n Step 2: Re-estimate equations including residuals as explanatory variables: Y n ' % X n % ˆµ n % g ( n W n ' % S n % ĝ n % µ ( n

Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16)

Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) 1 2 Model Consider a system of two regressions y 1 = β 1 y 2 + u 1 (1) y 2 = β 2 y 1 + u 2 (2) This is a simultaneous equation model