4. Standard Regression Model and Spatial Dependence Tests

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1 4. Stadard Regresso Model ad Spatal Depedece Tests Stadard regresso aalss fals the presece of spatal effects. I case of spatal depedeces ad/or spatal heterogeet a stadard regresso model wll be msspecfed. Spatal effects have to be corporated regresso models order to obta vald parameter estmates. Here we focus o spatal depedece of whch gorace causes severe terpretato problems ad requres a orgal spatal modellg approach. Spatal heterogeet ca much more be accouted for b methods developed mastream ecoometrcs. The stadard regresso model s usuall the startg pot of spatal regresso aalss. The resduals of ordar least-squares (OLS) estmato ca be used to test for spatal effects. Hece, we frst outle OLS estmato the stadard regresso model secto 4.. I secto 4. we troduce varous test for spatal effects the regresso model. Spatal regresso models suggested b the outcomes of the tests are preseted chapter 5.

2 4. The stadard regresso model Lear regresso model: Relatosshp betwee a depedet varable Y ad a set of explaator varables X, X,, X k. (4.) Xβ ε x vector of the depedet varable: xk matrx wth observatos of the k explaator varables: X x j : observato of the jth varable at the th statstcal ut st colum of X: vector of oes (for tercept) x x x... x x x k k k ' The explaator varables are treated as fxed ad ot radom. kx vector of regresso coeffcets: β k x vector of dsturbaces (error terms): ε ' '

3 Stadard assumptos:. The dsturbace has a expecto of zero: E( ) 0 for all. The dsturbaces have a costat varace (homoscedastct): Var ( ) E( ) for all, : error varace 3. The dsturbaces are ucorrelated (lack of autocorrelato): Cov(, j ) E( j ) 0 for all j Assumptos -3 compact form: E( ε) o ad Cov( ε) E( ε ε') o: x vector of zeros, I: x dett matrx For carrg out statstcal tests ormalt of the errors s assumed: I ~ N(0, ) for all or ~ N( o, I ) 3

4 Ordar least squares (OLS) estmato A mportat task of regresso aalss s to estmate the ukow vector of regresso coeffcets, β, order to assess the fluece of the regressors X, X,, X k o the depedet varable Y. Uder the stadard assumptos, ordar least squares (OLS) estmato elds best lear ubased estmators (blue propert). Least squares crtero: (4.a) Q( β) ε' ε ( Xβ)'( Xβ). Q has to be mmzed wth respect to β for whch we use the equvalet expresso: (4.b) Q( β ) ' ' Xβ β' X' Xβ Frst order codto for a mmum of Q: dq( β) dβ OLS estmator of β; X' ( X' X) β o (4.3) β ( X' X) X' 4

5 5 Ftted values, resduals ad resdual varace Ftted values: (4.4) Resduals: (4.5a) or (4.5b) Resdual varace (ubased estmate of ): (4.6) ( ) β X e β X e e 0 k ' e k e e Stadard error of regresso (SER): (4.7) SER

6 Measures of ft Decomposto of the total sum of squares of the depedet varable Y: (4.8) SST = SSE + SSR Total sum of squares: (4.9) SST ( ) Explaed sum of squares: (4.0) SSE (ŷ ) Resdual sum of squares: (4.) SSR ( ŷ) Coeffcet of determato: (4.a) R SSE SST Rage of R : 0 R or (4.b) e e'e R SSR SST 6

7 Adjusted coeffcet of determato: B the adjustmet regresso models wth dfferet umbers of regressors are made comparable. (4.3) R ( R ) k Iformato crtera Iformato measure the goodess of ft where model complext terms of the umber of explaator varables s pealzed. Goodess of ft s covered b the log lkelhood fucto l L, l( (4.4) l L C ), whch s mal composed of the sum of squared resduals. B pealzg fts wth a larger umber of regressors, regresso models wth dfferet k are made comparable. Accordg to the for-mato crtera, the model wth the lowest value s the best. - Akake formato crtero (AIC) - Schwartz crtero (SC) (4.5a) AIC = -l(l) + k (4.5b) SC = -l(l) + kl() 7

8 Hpothess tests -Test of sgfcace of regresso coeffcets Null hpothess H 0 : β j = 0 Dstrbuto of the OLS estmator β uder H 0 for ormall dstrbuted errors: β ~ N( β, ( X' X) ) because of Test statstc: (4.6) t j j xx jj xx jj : jth ma dagoal elemet of the verse (X X) - t j follows a t dstrbuto wth -k degrees of freedom. Sgfcace level: α Crtcal value (two-sded test): t(-k;-α/) E() β β ad Cov( β) σ ( X' X) Testg decso: t j > t(-k;-α/) => Reject H 0 or p < α => Reject H 0 p-value: Probablt of obtag a (absolute) hgher t statstc tha tj 8

9 - F test for the regresso as a whole Null hpothess H 0 : β = β 3 = = β k = 0 SSR c : Costraed resdual sum of squares from a regresso whch H 0 holds.e. a regresso of Y o the costat term X ol SSR u : Ucostraed resdual sum of squares from a regresso of Y o X, X,, X k Test statstc: (4.7a) or (4.7b) F (SSR c SSR u ) /(k SSR /( k) R F ( R u /(k ) ) /( k) ) F follows a F dstrbuto wth k- ad -k degrees of freedom. Testg decso: F > F(k-;-k;-α) => Reject H 0 or p < α => Reject H 0 9

10 Example: For 5 regos are data avalable o output growth (X) ad productvt growth (Y): Rego Output growth (X) Productvt growth (Y) Accordg to the Verdoor law output growth ad productvt growth are postvel related. Productvt growth creases wth output growth due to creasg returs to scale. The regresso model mpled b Verdoor s law reads (4.8) x wth x = for all ad x = x. If Verdoor s law holds, the Verdoor coeffcet β s expected to take a postve sg. The tercept captures productvt growth evoked b autoomous techcal progress. The regresso model (4.7) ca be estmated b OLS. 0

11 Vector of the edogeous varable : Observato matrx X: X ' Matrx product X X, ts verse (X X) -, matrx product X : X'X , ( X' X), X' OLS estmator of β: β ( X' X) X'

12 Vector of ftted values : ŷ β X ŷ e Vector of resduals e:

13 Resdual varace : e' e Stadard error of regresso (SER): SER

14 Coeffcet of determato Workg table ( 0.84 ) ŷ ) ( ŷ ) ( ŷ SST = 0.450, SSE = 0.43, SSR = SST SSE = = R SSE SST or R SSR SST

15 Test of sgfcace of regresso coeffcets - for β (H 0 : β = 0) OLS estmator for β : 0.65 Test statstc: t xx Crtcal value (α=0.05, two-sded test): t(3,0.975) = 3.8 Testg decso: ( t =.779) < [t(3;0.975)=3.8] => Accept H 0 - for β (H 0 : β = 0) OLS estmator for β : Test statstc: t xx Crtcal value (α=0.05, two-sded test): t(3,0.975) = 3.8 Testg decso: ( t =5.643) > [t(3;0.975)=3.8] => Reject H 0 5

16 F test for the regresso as a whole Null hpothess H 0 : β = 0 (ol oe o costat exogeous varable) Costraed resdual sum of squares: SSR c = SST = Ucostraed resdual sum of squares: SSR u = SSR = Test statstc: F (SSR c SSR u )/(k ) SSR /( k) u ( )/( ) /(5 ) R /(k ) 0.94/( ) or F ( R ) /( k) ( 0.94) /(5 ) (The dfferece of both computatos of F are ol due to roudg errors.) Crtcal value(α=0.05): F(;3;0.95) = 0. Testg decso: (F=3.948) > [F(;3;0.95)=0.] => Reject H 0 6

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The