Multi-Equation Structural Models: Seemingly Unrelated Regression Models

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1 Chapter 15 Multi-Equation Structural Models: Seemingly Unrelated Regression Models

2 Section 15.1 Seemingly Unrelated Regression Models

3 Modeling Approaches Econometric (Structural) Models Time-Series Models Intuitive Models Single- Equation Multi- Equation Univariate Multivariate Univariate Simultaneous Models Recursive Models Seemingly Unrelated Regression Models Box- Jenkins ARIMA Models Error Correction Models Vector Autoregression (VAR) Models Seasonal Smoothing Models Models Trend Extrapolation Models 3 In some instances, econometric models might encompass elements of timeseries models (for example, ARCH and GARCH models, adjustments for serial correlation).

4 Seemingly Unrelated Regression Models (SUR) Y Y : Y 1t 2t mt = = = β β 11 β 21 X m1 X 1t,1 X 2t,1 mt,1 + β 12 + β 22 + β X m2 1t,2 X 2t,2 X β mt,2 1k β X 2k β 1t, k1 X mkm + ε 2t, k 2 X 1t + ε mt, km 2t + ε SUR parameter estimates are full-information coefficients, whereas OLS parameter estimates are limited-information coefficients. mt Σ = σ σ m σ... σ 1m mm diagonal elements are variances; off-diagonal elements are covariances. 4 (1) If the RHS variables are the same across the m equations, the SUR parameter estimates are the same as the OLS parameter estimates even if Σ is not a diagonal matrix. (2) If the RHS variables are not the same across the m equations and if Σ is a diagonal matrix, then the SUR parameter estimates are the same as the OLS parameter estimates.

5 Section 15.2 Example of Seemingly Unrelated Regression Models

6 Example of Seemingly Unrelated Regression Models (Zellner 1962) I tm β β β + ε = 0m + 1mCt 1 + 2mFt 1 mt I = Gross Investment C = End-of-Period Capital Stock F = End-of-Period Value of Outstanding Shares Annual Data for Two Firms General Electric (GE) and Westinghouse (W) continued...

7 Σ is not a diagonal matrix represents the correlation of the disturbance terms in the GE and W equations. OLS SUR GE I W I t GE I W I t t = ( ) = t (8.0153) = ( ) = (7.5452) C t 1 (0.0257) C t 1 (0.0561) C C t 1 (0.0250) (0.053) t F t 1 (0.0156) F t 1 (0.0157) F F t 1 (0.0145) t 1 (0.0145) (standard errors in parentheses) 7 The use of SUR results in lower standard errors vis-àvis the use of OLS.

8 Section 15.3 Example: Demand for a Cereal Product from Five Retailers: HEB, Publix, Food Lion, Fred Myer, and Meijer

9 Example: Demand for a Cereal Product from Five Retailers: HEB, Publix, Food Lion, Fred Myer, and Meijer (No Adjustment for Serial Correlation) 9 SAS Program for Proc Syslin Use of Seemingly Unrelated Regression data HEB; input heb_retailer product week heb_logdisc heb_logprice heb_fsi1 weekno heb_logunits heb_logdisp heb_logad heb_logdist q1 q2 q3 q4; datalines; run; data PUBLIX; input publix_retailer product week publix_logdisc publix_logprice publix_fsi1 weekno publix_logunits publix_logdisp publix_logad publix_logdist q1 q2 q3 q4; datalines; run; data FOODLION; input flion_retailer product week flion_logdisc flion_logprice flion_fsi1 weekno flion_logunits flion_logdisp flion_logad flion_logdist q1 q2 q3 q4; datalines; run;

10 data FREDMYER; input fmyer_retailer product week fmyer_logdisc fmyer_logprice fmyer_fsi1 weekno fmyer_logunits fmyer_logdisp fmyer_logad fmyer_logdist q1 q2 q3 q4; datalines; run; data MEIJER; input meijer_retailer product week meijer_logdisc meijer_logprice meijer_fsi1 weekno meijer_logunits meijer_logdisp meijer_logad meijer_logdist q1 q2 q3 q4; datalines; run; options nodate; data all; merge heb publix foodlion fredmyer meijer; by week; proc means data=all n mean median std min max; var heb_logunits publix_logunits flion_logunits fmyer_logunits meijer_logunits; run; proc means data=all n mean median std min max; var heb_logprice publix_logprice flion_logprice fmyer_logprice meijer_logprice; run; 10

11 proc syslin data=all sur out=retailersur; heb: model heb_logunits=heb_logprice heb_logdisc heb_fsi1 heb_logdisp heb_logad heb_logdist weekno q1 q2 q3 / dw; output predicted=pheb_logunits; publix: model publix_logunits=publix_logprice publix_logdisc publix_fsi1 publix_logdisp publix_logad publix_logdist weekno q1 q2 q3 / dw; output predicted=ppublix_logunits; flion: model flion_logunits=flion_logprice flion_logdisc flion_fsi1 flion_logdisp flion_logad flion_logdist weekno q1 q2 q3 / dw; output predicted=pflion_logunits; fmyer: model fmyer_logunits=fmyer_logprice fmyer_logdisc fmyer_fsi1 fmyer_logdisp fmyer_logad fmyer_logdist weekno q1 q2 q3 / dw; output predicted=pfmyer_logunits; meijer: model meijer_logunits=meijer_logprice meijer_logdisc meijer_fsi1 meijer_logdisp meijer_logad meijer_logdist weekno q1 q2 q3 / dw; output predicted=pmeijer_logunits; 11

12 stest heb.heb_logprice-publix.publix_logprice=0; stest heb.heb_logprice-flion.flion_logprice=0; stest heb.heb_logprice-fmyer.fmyer_logprice=0; stest heb.heb_logprice-meijer.meijer_logprice=0; stest publix.publix_logprice-flion.flion_logprice=0; stest publix.publix_logprice-fmyer.fmyer_logprice=0; stest publix.publix_logprice-meijer.meijer_logprice=0; stest flion.flion_logprice-fmyer.fmyer_logprice=0; stest flion.flion_logprice-meijer.meijer_logprice=0; stest fmyer.fmyer_logprice-meijer.meijer_logprice=0; run; System tests of the own-price elasticities 12

13 proc corr data=retailersur; var heb_logunits pheb_logunits; run; proc corr data=retailersur; var publix_logunits ppublix_logunits; run; proc corr data=retailersur; var flion_logunits pflion_logunits; run; proc corr data=retailersur; var fmyer_logunits pfmyer_logunits; run; proc corr data=retailersur; var meijer_logunits pmeijer_logunits; run; 13

14 The MEANS Procedure Variable N Mean Median Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ heb_logunits publix_logunits flion_logunits fmyer_logunits meijer_logunits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ The MEANS Procedure Variable N Mean Median Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ heb_logprice publix_logprice flion_logprice fmyer_logprice meijer_logprice ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 14

15 The SYSLIN Procedure Ordinary Least Squares Estimation Model HEB Dependent Variable heb_logunits Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

16 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 heb_logprice heb_logdisc heb_fsi heb_logdisp <.0001 heb_logad heb_logdist <.0001 weekno q q q

17 The SYSLIN Procedure Ordinary Least Squares Estimation Model PUBLIX Dependent Variable publix_logunits Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

18 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 publix_logprice <.0001 publix_logdisc <.0001 publix_fsi publix_logdisp <.0001 publix_logad <.0001 publix_logdist weekno q q q

19 The SYSLIN Procedure Ordinary Least Squares Estimation Model FLION Dependent Variable flion_logunits Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

20 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept flion_logprice flion_logdisc <.0001 flion_fsi flion_logdisp <.0001 flion_logad flion_logdist <.0001 weekno q q q

21 The SYSLIN Procedure Ordinary Least Squares Estimation Model FMYER Dependent Variable fmyer_logunits Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

22 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 fmyer_logprice <.0001 fmyer_logdisc <.0001 fmyer_fsi fmyer_logdisp <.0001 fmyer_logad fmyer_logdist weekno q q q

23 The SYSLIN Procedure Ordinary Least Squares Estimation Model MEIJER Dependent Variable meijer_logunits Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

24 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept meijer_logprice meijer_logdisc <.0001 meijer_fsi meijer_logdisp <.0001 meijer_logad meijer_logdist weekno q q q

25 The SYSLIN Procedure Seemingly Unrelated Regression Estimation Cross Model Covariance HEB PUBLIX FLION FMYER MEIJER HEB PUBLIX FLION FMYER MEIJER Cross Model Correlation Σ matrix in correlation form HEB PUBLIX FLION FMYER MEIJER 25 HEB PUBLIX FLION FMYER MEIJER

26 Cross Model Inverse Correlation HEB PUBLIX FLION FMYER MEIJER HEB PUBLIX FLION FMYER MEIJER Cross Model Inverse Covariance HEB PUBLIX FLION FMYER MEIJER HEB PUBLIX FLION FMYER MEIJER System Weighted MSE Degrees of freedom 750 System Weighted R-Square

27 The SYSLIN Procedure Seemingly Unrelated Regression Estimation Model HEB Dependent Variable heb_logunits Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t 27 Intercept <.0001 heb_logprice heb_logdisc heb_fsi heb_logdisp <.0001 heb_logad heb_logdist <.0001 weekno q q q

28 Durbin-Watson Number of Observations 161 First-Order Autocorrelation Model PUBLIX Dependent Variable publix_logunits Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 publix_logprice <.0001 publix_logdisc <.0001 publix_fsi publix_logdisp <.0001 publix_logad <.0001 publix_logdist weekno q q q

29 The SYSLIN Procedure Seemingly Unrelated Regression Estimation Durbin-Watson Number of Observations 161 First-Order Autocorrelation Model FLION Dependent Variable flion_logunits Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t 29 Intercept flion_logprice flion_logdisc <.0001 flion_fsi flion_logdisp <.0001 flion_logad flion_logdist <.0001 weekno q q q

30 Durbin-Watson Number of Observations 161 First-Order Autocorrelation Model FMYER Dependent Variable fmyer_logunits Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 fmyer_logprice <.0001 fmyer_logdisc <.0001 fmyer_fsi fmyer_logdisp <.0001 fmyer_logad fmyer_logdist weekno q q q

31 Durbin-Watson Number of Observations 161 First-Order Autocorrelation Model MEIJER Dependent Variable meijer_logunits 31 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 meijer_logprice meijer_logdisc <.0001 meijer_fsi meijer_logdisp <.0001 meijer_logad meijer_logdist weekno q q q

32 Durbin-Watson Number of Observations 161 First-Order Autocorrelation Test Results of Hypotheses Regarding Own-Price Elasticites Num DF Den DF F Value Pr > F (1) H 0 : HEB = PUBLIX (2) (3) H 0 : HEB = FOOD LION H 0 : HEB = FRED MYER (4) H 0 : HEB = MEIJER (5) (6) H 0 : PUBLIX = FOOD LION H 0 : PUBLIX = FRED MYER 32

33 Num DF Den DF F Value Pr > F (7) H 0 : PUBLIX = MEIJER (8) H 0 : FOOD LION = FRED MYER (9) H 0 : FOOD LION = MEIJER (10) H 0 : FRED MYER = MEIJER 33

34 The CORR Procedure 2 Variables: heb_logunits pheb_logunits Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum heb_logunits pheb_logunits Simple Statistics Variable Label heb_logunits pheb_logunits Predicted Values 34

35 Pearson Correlation Coefficients, N = 164 Prob > r under H0: Rho=0 heb_ pheb_ logunits logunits heb_logunits <.0001 pheb_logunits Predicted Values <.0001 R 2 = (.83765) 2 = The CORR Procedure 2 Variables: publix_logunits ppublix_logunits Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum publix_logunits ppublix_logunits

36 Simple Statistics Variable Label publix_logunits ppublix_logunits Predicted Values Pearson Correlation Coefficients, N = 165 Prob > r under H0: Rho=0 publix_ ppublix_ logunits logunits publix_logunits <.0001 ppublix_logunits Predicted Values < R 2 = (.97490) 2 =

37 The CORR Procedure 2 Variables: flion_logunits pflion_logunits Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum flion_logunits pflion_logunits Simple Statistics Variable Label flion_logunits pflion_logunits Predicted Values 37

38 Pearson Correlation Coefficients, N = 165 Prob > r under H0: Rho=0 flion_ pflion_ logunits logunits flion_logunits <.0001 pflion_logunits Predicted Values <.0001 R 2 = (.98733) 2 =

39 The CORR Procedure 2 Variables: fmyer_logunits pfmyer_logunits Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum fmyer_logunits pfmyer_logunits Simple Statistics Variable Label fmyer_logunits pfmyer_logunits Predicted Values 39

40 Pearson Correlation Coefficients, N = 165 Prob > r under H0: Rho=0 fmyer_ pfmyer_ logunits logunits fmyer_logunits <.0001 pfmyer_logunits Predicted Values <.0001 R 2 = (.94580) 2 =

41 The CORR Procedure 2 Variables: meijer_logunits pmeijer_logunits Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum meijer_logunits pmeijer_logunits Simple Statistics Variable Label meijer_logunits pmeijer_logunits Predicted Values 41

42 Pearson Correlation Coefficients, N = 162 Prob > r under H0: Rho=0 meijer_ pmeijer_ logunits logunits meijer_logunits <.0001 pmeijer_logunits Predicted Values <.0001 R 2 = (.96525) 2 =

43 Use of Proc Model: SUR Analysis With and Without Adjustment for Serial Correlation options nodate; data all; merge heb publix foodlion fredmyer meijer; by week; proc means data=all n mean median std min max; var heb_logunits publix_logunits flion_logunits fmyer_logunits meijer_logunits; run; proc means data=all n mean median std min max; var heb_logprice publix_logprice flion_logprice fmyer_logprice meijer_logprice; run; * without adjustment for serial correlation; 43

44 proc model data=all; heb_logunits=a0+a1*heb_logprice+a2*heb_logdisc+a3*heb_fsi1+ a4*heb_logdisp+a5*heb_logad+a6*heb_logdist+ a7*weekno+a8*q1+a9*q2+a10*q3; publix_logunits=b0+b1*publix_logprice+b2*publix_logdisc+ b3*publix_fsi1+b4*publix_logdisp+ b5*publix_logad+b6*publix_logdist+b7*weekno+b8*q1+b9*q2+b10*q3; flion_logunits=c0+c1*flion_logprice+c2*flion_logdisc+c3*flion_fsi1+ c4*flion_logdisp+c5*flion_logad+ c6*flion_logdist+c7*weekno+c8*q1+c9*q2+c10*q3; fmyer_logunits=d0+d1*fmyer_logprice+d2*fmyer_logdisc+d3*fmyer_fsi1+ d4*fmyer_logdisp+d5*fmyer_logad+ d6*fmyer_logdist+d7*weekno+d8*q1+d9*q2+d10*q3; meijer_logunits=e0+e1*meijer_logprice+e2*meijer_logdisc+ e3*meijer_fsi1+e4*meijer_logdisp+e5*meijer_logad+ e6*meijer_logdist+e7*weekno+e8*q1+e9*q2+e10*q3; 44

45 fit heb_logunits publix_logunits flion_logunits fmyer_logunits meijer_logunits / sur dw dwprob normal out=retailersur; parms a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 c0 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10; test a1-b1=0; test a1-c1=0; test a1-d1=0; test a1-e1=0; test b1-c1=0; test b1-d1=0; test b1-e1=0; test c1-d1=0; test c1-e1=0; test d1-e1=0; run; * with adjustment for serial correlation; 45

46 46 proc model data=all; heb_logunits=a0+a1*heb_logprice+a2*heb_logdisc+a3* heb_fsi1+a4*heb_logdisp+a5*heb_logad+a6*heb_logdist+ a7*weekno+a8*q1+a9*q2+a10*q3; publix_logunits=b0+b1*publix_logprice+b2*publix_logdisc+ b3*publix_fsi1+b4*publix_logdisp+ b5*publix_logad+b6*publix_logdist+b7*weekno+b8*q1+ b9*q2+b10*q3; flion_logunits=c0+c1*flion_logprice+c2*flion_logdisc+ c3*flion_fsi1+c4*flion_logdisp+c5*flion_logad+ c6*flion_logdist+c7*weekno+c8*q1+c9*q2+c10*q3; fmyer_logunits=d0+d1*fmyer_logprice+d2*fmyer_logdisc+ d3*fmyer_fsi1+d4*fmyer_logdisp+d5*fmyer_logad+ d6*fmyer_logdist+d7*weekno+d8*q1+d9*q2+d10*q3; meijer_logunits=e0+e1*meijer_logprice+e2*meijer_logdisc+ e3*meijer_fsi1+e4*meijer_logdisp+e5*meijer_logad+ e6*meijer_logdist+e7*weekno+e8*q1+e9*q2+e10*q3; %ar(heb_logunits,1); %ar(publix_logunits,1); %ar(flion_logunits,1); %ar(fmyer_logunits,1); %ar(meijer_logunits,1); AR(1) process in each of the five equations which comprise the system of seemingly unrelated regressions

47 fit heb_logunits publix_logunits flion_logunits fmyer_logunits meijer_logunits / sur dw dwprob normal out=retailersur; parms a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 c0 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 d0 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10; test a1-b1=0; test a1-c1=0; test a1-d1=0; test a1-e1=0; test b1-c1=0; test b1-d1=0; test b1-e1=0; test c1-d1=0; test c1-e1=0; test d1-e1=0; run; 47

48 Use of PROC Model to Obtain Estimates of Parameters from a System of Seemingly Unrelated Regressions The MODEL Procedure Model Summary Model Variables 5 Parameters 55 Equations 5 Number of Statements 25 The 5 Equations to Estimate 48

49 heb_logunits = F(a0(1), a1(heb_logprice), a2(heb_logdisc), a3(heb_fsi1), a4(heb_logdisp), a5(heb_logad), a6(heb_logdist), a7(weekno), a8(q1), a9(q2), a10(q3)) publix_logunits = F(b0(1), b1(publix_logprice), b2(publix_logdisc), b3(publix_fsi1), b4(publix_logdisp), b5(publix_logad), b6(publix_logdist), b7(weekno), b8(q1), b9(q2), b10(q3)) flion_logunits = F(c0(1), c1(flion_logprice), c2(flion_logdisc), c3(flion_fsi1), c4(flion_logdisp), c5(flion_logad), c6(flion_logdist), c7(weekno), c8(q1), c9(q2), c10(q3)) fmyer_logunits = F(d0(1), d1(fmyer_logprice), d2(fmyer_logdisc), d3(fmyer_fsi1), d4(fmyer_logdisp), d5(fmyer_logad), d6(fmyer_logdist), d7(weekno), d8(q1), d9(q2), d10(q3)) meijer_logunits = F(e0(1), e1(meijer_logprice), e2(meijer_logdisc), e3(meijer_fsi1), e4(meijer_logdisp), e5(meijer_logad), e6(meijer_logdist), e7(weekno), e8(q1), e9(q2), e10(q3)) 49 NOTE: At SUR Iteration 1 CONVERGE=0.001 Criteria Met.

50 The MODEL Procedure SUR Estimation Summary Data Set Options DATA= ALL OUT= RETAILERSUR Minimization Summary Parameters Estimated 55 Method Gauss Iterations 1 Final Convergence Criteria R 0 PPC 5.64E-11 RPC(e7) Object Trace(S) Objective Value Observations Processed 50 Read 165 Solved 165 Used 161 Missing 4

51 Number of parameters to estimate Difference between number of observations and number of parameters to estimate The MODEL Procedure Nonlinear SUR Summary of Residual Errors DF DF Adj Durbin Equation Model Error SSE MSE Root MSE R-Square R-Sq Watson heb_logunits publix_logunits flion_logunits fmyer_logunits meijer_logunits parameters to estimate, including the intercept 51

52 Nonlinear SUR Parameter Estimates Approx Approx Parameter Estimate Std Err t Value Pr > t 52 a <.0001 a a a a <.0001 a a <.0001 a a a a b <.0001 b <.0001 b <.0001 b b <.0001 b <.0001 b b7 Precisely the same estimates and standard b errors obtained with Proc SYSLIN b b

53 c c c <.0001 c c <.0001 c c <.0001 c c c c d <

54 d <.0001 d <.0001 d d <.0001 d d d d d d e <.0001 e e <.0001 e e <.0001 e e e e e e

55 Test Results SUR results no adjustment for serial correlation Test Type Statistic Pr > ChiSq Label Test0 Wald <.0001 a1-b1=0 Test1 Wald a1-c1=0 Test2 Wald <.0001 a1-d1=0 Test3 Wald a1-e1=0 Test4 Wald b1-c1=0 Test5 Wald b1-d1=0 Test6 Wald b1-e1=0 Test7 Wald c1-d1=0 Test8 Wald c1-e1=0 Test9 Wald d1-e1=0 Rejection of hypotheses Number of Observations Statistics for System 55 Used 161 Objective Missing 4 Objective*N

56 The MODEL Procedure Durbin-Watson Statistics Equation Order DW Pr < DW Pr > DW heb_logunits < publix_logunits < flion_logunits fmyer_logunits < meijer_logunits Evidence of AR(1) process in all equations. 56

57 Evidence concerning normality of residuals for individual equations and for the system as a whole except for the food lion equation; normality of residuals is evident in heb, publix, fred myer, and meijer equations. Normality Test Equation Test Statistic Value Prob heb_logunits Shapiro-Wilk W publix_logunits Shapiro-Wilk W flion_logunits Shapiro-Wilk W fmyer_logunits Shapiro-Wilk W meijer_logunits Shapiro-Wilk W System Mardia Skewness <.0001 Mardia Kurtosis 9.57 <.0001 Henze-Zirkler T For the system as a whole, normality of residuals is evident.

58 The NORMAL option in the FIT statement of Proc Model allows the construction of multivariate and univariate tests of normality of residuals. Shipiro-Wilk W test: a univariate test, one for each equation. The W test is performed because the sample size is less than 2,000 observations. Computation of the Kolmogorov-Smirnoff test statistic (the other univariate test associated with Proc Model) requires at least 2,000 observations. Mandia Skewness (1980): a test of multivariate skewness 2 (asymptotically distributed as χ ). Mandia Kurtosis: a test of multivariate Kurtosis (asymptotically distributed as normal). Henze-Zirkler T-statistic (1990): a test of multivariate normality (asymptotically distributed as log normal). 58

59 The MODEL Procedure SUR Estimation Summary Data Set Options DATA= ALL OUT= RETAILERSUR Minimization Summary Parameters Estimated 60 Method Gauss Iterations 4 Final Convergence Criteria R PPC(e3) RPC(e3) Object 2.727E-6 Trace(S) Objective Value Observations Processed 59 Read 165 Solved 165 Used 161 Missing 4

60 The MODEL Procedure Nonlinear SUR Summary of Residual Errors DF DF Adj Durbin Equation Model Error SSE MSE Root MSE R-Square R-Sq Watson heb_logunits publix_logunits flion_logunits fmyer_logunits meijer_logunits * With adjustment for serial correlation 60

61 Nonlinear SUR Parameter Estimates Approx Approx Parameter Estimate Std Err t Value Pr > t Label 61 a <.0001 a a a a <.0001 a a a a a a b <.0001 b b <.0001 b b <.0001 b <.0001 b b b b b

62 62 c c c <.0001 c c <.0001 c c <.0001 c c c c d <.0001 d <.0001 d <.0001 d d <.0001 d d d d d d

63 e e e <.0001 e e <.0001 e e e e e e heb_logunits_l <.0001 AR(heb_logunits) heb_logunits lag1 parameter publix_logunits_l <.0001 AR(publix_logunits) publix_logunits lag1 parameter flion_logunits_l AR(flion_logunits) flion_logunits lag1 parameter fmyer_logunits_l <.0001 AR(fmyer_logunits) fmyer_logunits lag1 parameter meijer_logunits_l AR(meijer_logunits) meijer_logunits lag1 parameter 63

64 Test Results (with adjustment for serial correlation) Test Type Statistic Pr > ChiSq Label Test0 Wald a1-b1=0 Test1 Wald a1-c1=0 Test2 Wald a1-d1=0 Test3 Wald a1-e1=0 Test4 Wald b1-c1=0 Test5 Wald b1-d1=0 Test6 Wald b1-e1=0 Test7 Wald c1-d1=0 Test8 Wald c1-e1=0 Test9 Wald d1-e1=0 Number of Observations Statistics for System 64 Used 161 Objective Missing 4 Objective*N

65 The MODEL Procedure Durbin-Watson Statistics Equation Order DW Pr < DW Pr > DW heb_logunits publix_logunits flion_logunits fmyer_logunits meijer_logunits Serial correlation no longer a problem in any equation. 65

66 Normality Test Equation Test Statistic Value Prob heb_logunits Shapiro-Wilk W publix_logunits Shapiro-Wilk W flion_logunits Shapiro-Wilk W fmyer_logunits Shapiro-Wilk W 0.95 <.0001 meijer_logunits Shapiro-Wilk W System Mardia Skewness <.0001 Mardia Kurtosis <.0001 Henze-Zirkler T Non-normality of residuals in heb, flion, and fmyer equations. Non-normality of residuals as a system. 66

67 Variable SUR Results Demand for a Cereal Product : H-E-B Parameter Estimate OLS SUR a SUR b Standard Error Parameter Estimate Standard Error Parameter Estimate Standard Error Intercept log price log DISC Fsi log DISP log AD log DIST week no q q q R c (0.7086) d e AR(1)

68 SUR Results Demand for a Cereal Product : PUBLIX Variable Parameter Estimate OLS SUR a SUR b Standard Error Parameter Estimate Standard Error Parameter Estimate Standard Error Intercept log price log DISC Fsi log DISP log AD log DIST week no q q q R c d e AR(1)

69 Variable SUR Results Demand for a Cereal Product : Food OLS SUR Lion a SUR b Parameter Estimate Standard Error Parameter Estimate Standard Error Parameter Estimate Standard Error Intercept log price log DISC Fsi log DISP log AD log DIST week no q q q R c (0.9752) d e AR(1)

70 SUR Results Demand for a Cereal Product : FRED MYER Variable Parameter Estimate OLS SUR a SUR b Standard Error Parameter Estimate Standard Error Parameter Estimate Standard Error Intercept log price log DISC Fsi log DISP log AD log DIST week no q q q R c d e AR(1)

71 SUR Results Demand for a Cereal Product : MEIJER Variable Parameter Estimate OLS SUR a SUR b Standard Error Parameter Estimate Standard Error Parameter Estimate Standard Error Intercept log price log DISC Fsi log DISP log AD log DIST week no Q Q q R c d e AR(1)

72 Section 15.4 Seemingly Unrelated Regression Models with Restrictions

73 Often models require restrictions to conform to theoretical developments. Consumer demand functions derived from neoclassical theory possess the mathematical properties of homogeneity, adding-up, symmetry, and negativity. When estimating demand models, economists often impose homogeneity and symmetry as parameter restrictions. Statistical rejection of theoretical restrictions is NOT unusual; however, it is commonplace to see theoretical restrictions imposed in spite of their statistical failure. 73

74 Section 15.5 Rotterdam Model

75 Rotterdam Model (Barten, 1965) 75 w * it Dq Dq it it where : Dp Dy w w y * it it t t jt = = = = = = = b i Dy [ q / q ] log it it p i log log it t k [ p / p ] [ y / y ] ( w q p it it t it / q it jt + y t t 1 w jt 1 it 1 ) w * kt Dp kt + j c ij Dp jt + ε RESTRICTIONS c ij = 0 j c = c = total expenditure at time period t. ij ji it (HOMOGENEITY) (SYMMETRY) This model involves logarithmic differences of quantities, prices, and total expenditure.

76 Section 15.6 Linear Approximate Almost Ideal Demand System (LA/AIDS) Model

77 Linear Approximate (LA) Almost Ideal Demand System (AIDS) (Deaton and Muellbauer, 1980) w it = α + i j γ ij log p jt + β log( y i t / P t ) + ε it log P t = k w kt log p kt (Stone Price Index) RESTRICTIONS j γ ij = 0 (HOMOGENEITY) 77 γ ij = γ ji (SYMMETRY) A very popular model among demand economists.

78 Section 15.7 Example: Demand Interrelationships for Spaghetti Sauces: LA/AIDS Model

79 Example: Demand Interrelationships for Spaghetti Sauces: Use of LA/AIDS Model with Restrictions Use of PROC SYSLIN * Demonstration of Seemingly Unrelated Regression Model with Restrictions; * Demand Analysis of Various Spaghetti Sauces; data spaghettisauce; input WEEK QCLASSICO PCLASSICO QHUNTS PHUNTS QNEWMAN PNEWMAN QPREGO PPREGO QPRIVLABEL PPRIVLABEL QRAGU PRAGU TOTALEXP; wclassico=qclassico*pclassico/totalexp; whunts=qhunts*phunts/totalexp; wnewman=qnewman*pnewman/totalexp; wprego=qprego*pprego/totalexp; wprivlabel=qprivlabel*pprivlabel/totalexp; wragu=qragu*pragu/totalexp; 79

80 wcheck=wclassico+whunts+wnewman+wprego+ wprivlabel+wragu; lpclassico=log(pclassico); lphunts=log(phunts); lpnewman=log(pnewman); lpprego=log(pprego); lpprivlabel=log(pprivlabel); lpragu=log(pragu); StonePI=wclassico*lpclassico+whunts*lphunts+ wnewman*lpnewman+wprego*lpprego+ wprivlabel*lpprivlabel+wragu*lpragu; ltotexp=log(totalexp); lrealexp=ltotexp-stonepi; datalines; 80

81 options nodate; proc means data=spaghettisauce n mean median std min max; var QCLASSICO QHUNTS QNEWMAN QPREGO QPRIVLABEL QRAGU; run; proc means data=spaghettisauce n mean median std min max; var PCLASSICO PHUNTS PNEWMAN PPREGO PPRIVLABEL PRAGU; run; proc means data=spaghettisauce n mean median std min max; var WCLASSICO WHUNTS WNEWMAN WPREGO WPRIVLABEL WRAGU wcheck; run; 81

82 82 * Seemingly Unrelated Regression LA/AIDS Model; proc syslin data=spaghettisauce itsur; classico: model wclassico=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; hunts: model whunts=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; newman: model wnewman=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; ragu: model wragu=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; prego: model wprego=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw;

83 Homogeneity Restrictions srestrict classico.lpclassico+classico.lphunts+ classico.lpnewman+classico.lpprego+classico. lpprivlabel+classico.lpragu=0, hunts.lpclassico+hunts.lphunts+hunts.lpnewman+hunts. lpprego+hunts.lpprivlabel+hunts.lpragu=0, newman.lpclassico+newman.lphunts+newman.lpnewman+ newman.lpprego+newman.lpprivlabel+newman.lpragu=0, ragu.lpclassico+ragu.lphunts+ragu.lpnewman+ragu. lpprego+ragu.lpprivlabel+ragu.lpragu=0, prego.lpclassico+prego.lphunts+prego.lpnewman+prego. lpprego+prego.lpprivlabel+prego.lpragu=0, 83

84 Symmetry Restrictions classico.lphunts-hunts.lpclassico=0, classico.lpnewman-newman.lpclassico=0, classico.lpragu-ragu.lpclassico=0, classico.lpprego-prego.lpclassico=0, hunts.lpnewman-newman.lphunts=0, hunts.lpragu-ragu.lphunts=0, hunts.lpprego-prego.lphunts=0, newman.lpragu-ragu.lpnewman=0, newman.lpprego-prego.lpnewman=0, ragu.lpprego-prego.lpragu=0; run; 84

85 NOTE: Due to adding-up, one equation must be dropped to avoid the singularity of Σ. In this case, the omitted equation is for private label spaghetti sauce. * system test of homogeneity and symmetry restrictions; proc syslin data=spaghettisauce itsur; classico: model wclassico=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; hunts: model whunts=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; newman: model wnewman=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; ragu: model wragu=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; prego: model wprego=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; 85

86 Test of homogeneity restrictions stest classico.lpclassico+classico.lphunts+classico. lpnewman+classico.lpprego+classico.lpprivlabel+ classico.lpragu=0, hunts.lpclassico+hunts.lphunts+hunts.lpnewman+hunts. lpprego+hunts.lpprivlabel+hunts.lpragu=0, newman.lpclassico+newman.lphunts+newman.lpnewman+newman. lpprego+newman.lpprivlabel+newman.lpragu=0, ragu.lpclassico+ragu.lphunts+ragu.lpnewman+ragu.lpprego+ ragu.lpprivlabel+ragu.lpragu=0, prego.lpclassico+prego.lphunts+prego.lpnewman+prego. lpprego+prego.lpprivlabel+prego.lpragu=0, 86

87 Test of symmetry restrictions classico.lphunts-hunts.lpclassico=0, classico.lpnewman-newman.lpclassico=0, classico.lpragu-ragu.lpclassico=0, classico.lpprego-prego.lpclassico=0, hunts.lpnewman-newman.lphunts=0, hunts.lpragu-ragu.lphunts=0, hunts.lpprego-prego.lphunts=0, newman.lpragu-ragu.lpnewman=0, newman.lpprego-prego.lpnewman=0, ragu.lpprego-prego.lpragu=0; run; * system test of homogeneity restrictions only; 87

88 proc syslin data=spaghettisauce itsur; classico: model wclassico=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; hunts: model whunts=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; newman: model wnewman=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; ragu: model wragu=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; prego: model wprego=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; stest 88

89 Test of homogeneity restrictions only classico.lpclassico+classico.lphunts+classico.lpnewman+ classico.lpprego+classico.lpprivlabel+classico.lpragu=0, hunts.lpclassico+hunts.lphunts+hunts.lpnewman+hunts.lpprego+ hunts.lpprivlabel+hunts.lpragu=0, newman.lpclassico+newman.lphunts+newman.lpnewman+newman. lpprego+newman.lpprivlabel+newman.lpragu=0, 89 ragu.lpclassico+ragu.lphunts+ragu.lpnewman+ragu.lpprego+ragu. lpprivlabel+ragu.lpragu=0, prego.lpclassico+prego.lphunts+prego.lpnewman+prego.lpprego+ prego.lpprivlabel+prego.lpragu=0; run; * system test of symmetry restrictions only;

90 proc syslin data=spaghettisauce itsur; classico: model wclassico=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; hunts: model whunts=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; newman: model wnewman=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; ragu: model wragu=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; prego: model wprego=lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp / dw; stest 90

91 Test of symmetry restrictions only classico.lphunts-hunts.lpclassico=0, classico.lpnewman-newman.lpclassico=0, classico.lpragu-ragu.lpclassico=0, classico.lpprego-prego.lpclassico=0, hunts.lpnewman-newman.lphunts=0, hunts.lpragu-ragu.lphunts=0, hunts.lpprego-prego.lphunts=0, newman.lpragu-ragu.lpnewman=0, newman.lpprego-prego.lpnewman=0, ragu.lpprego-prego.lpragu=0; run; 91

92 The MEANS Procedure Volume of Spaghetti Sauce in standard units (15 oz container size) Variable N Mean Median Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ QCLASSICO QHUNTS QNEWMAN QPREGO QPRIVLABEL QRAGU ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ The MEANS Procedure Prices/Standardized container Variable N Mean Median Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ PCLASSICO PHUNTS PNEWMAN PPREGO PPRIVLABEL PRAGU ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 92

93 The MEANS Procedure Market Shares Variable N Mean Median Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ wclassico whunts wnewman wprego wprivlabel wragu wcheck E ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Market shares need to add to 1. Ragu and Prego are the industry leaders 93

94 The SYSLIN Procedure Ordinary Least Squares Estimation (No Restrictions Imposed) Model CLASSICO Dependent Variable wclassico Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

95 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp <

96 The SYSLIN Procedure Ordinary Least Squares Estimation Model HUNTS Dependent Variable whunts Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

97 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept lpclassico lphunts <.0001 lpnewman lpprego lpprivlabel lpragu lrealexp

98 The SYSLIN Procedure Ordinary Least Squares Estimation Model NEWMAN Dependent Variable wnewman Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error E-6 Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

99 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 lpclassico lphunts lpnewman <.0001 lpprego lpprivlabel lpragu lrealexp

100 The SYSLIN Procedure Ordinary Least Squares Estimation Model RAGU Dependent Variable wragu Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

101 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept lpclassico lphunts lpnewman lpprego lpprivlabel lpragu lrealexp <

unadjusted model for baseline cholesterol 22:31 Monday, April 19,

unadjusted model for baseline cholesterol 22:31 Monday, April 19, 2004 1 Class Level Information Class Levels Values TRETGRP 3 3 4 5 SEX 2 0 1 Number of observations 916 unadjusted model for baseline cholesterol