CEU Departmet of Ecoomics Ecoometrics, Problem Set - Solutios Part A. Exogeeity - edogeeity The liear coditioal expectatio (CE) model has the followig form: We would like to estimate the effect of some variable, ω, o aother, y, holdig all other correlated with y variables, c, costat. Assumig that all variables are mea 0, there is o costat i the model. So, the structural model is: E( y ω, c) = αω + αc+ ε, where α, α are parameters ad ε is a error term. If ω ad c are correlated, the E( c ω)= γω ad the observable/estimable liear CE model is: E( y ω) = αω + αγω+ ε = ( α+ αγ ) ω+ ε = βω+ ε, where β = α+ αγ. Therefore, observig β we have a direct effect of ω o y (α ) plus a idirect effect (α γ ), which is the effect of c o y (α ) multiplied by the effect of ω o c (γ ). I the liear model the average partial effect (APE) equals the partial effect (PE) because the latter is costat: APE= PE= δ E( y ω, c) / δω = α. If we caot cotrol c ( α 0 ), ad if ω ad c are correlated ( γ 0 ), the the observable effect would be: δ E( y ω) / δω = β = α+ α γ, ad we caot idetify the PE of ω o y (α ). The bias would be: Bias= β α= α γ. y : uemploymet, ω : iflatio Estimatig the effect of iflatio o uemploymet we would expect: α < 0, i.e. the sig of the APE to be (-). There are may other variables that have effect o both, iflatio ad uemploymet, ad for this reaso we would like to cotrol them. Examples: GDP gap, i.e. the differece betwee the actual ad potetial gross domestic product, domestic demad, fiscal policy (both taxes ad Govermet expediture), level of educatio, literacy rate, etc. Especially importat variable i my view is the GDP gap (Yg). We would expect α < 0, i.e. whe the Yg icreases the uemploymet falls ad vice versa whe Yg decreases the uemploymet rises. The effect of Yg o iflatio is: γ > 0, i.e. whe Yg icreases the iflatio also icreases ad vice versa. Therefore, if we ca t measure Yg we ca t idetify the APE, ω is edogeous ad the bias would be egative.. y : GDP, ω : degree of opeess, e.g. low barriers to imports Other variables correlated with either ω or y, or both, are: populatio, workig force, domestic demad, political stability, fiacial stability, exchage rate, iterest rates or moey supply volatility. I would call my variable: moetary policy soudess, ad i my view it is correlated to both, ω ad y. The expected sigs of the parameters are: α> 0, α> 0, γ> 0, i.e. icreasig the degree of opeess would raise the GDP; icreasig the moetary policy soudess would icrease both, the GDP ad the degree of opeess. If we ca t measure c we ca t idetify the APE, ω is edogeous ad the bias would be positive.. y : employmet opportuities of ethic miorities, ω : ati-discrimiatio laws The itroductio of ati-discrimiatio laws (or the icrease of their amout) would raise the employmet opportuities of ethic miorities. Other variables that correlate with either ω or y, or both, are: uemploymet level, stage of ecoomic or political cycle, GDP gap. Importat variable i my view is the uemploymet. Icrease of uemploymet would promote the st year MA i Ecoomics
itroductio of ati-discrimiatio laws but would decrease the employmet opportuities of ethic miorities. If we ca t measure c the expected sigs of the parameters would be: α> 0, α < 0, γ > 0. Therefore, we ca t idetify the APE i that situatio, ω is edogeous ad the bias would be egative.. y : lifetime earigs, ω : geder Estimatig the relatioship betwee geder ad lifetime-earigs we would expect males to ear more tha females. Other variables that correlate with either ω or y, or both, are: educatio, abilities, age, occupatio. Pickig educatio as particularly importat variable we would expect that it would be positively correlated with life-time earigs ad (arguably) ucorrelated with geder. We could otherwise suppose that it is positively correlated with geder (if we substitute male= ad female=0). I the first case: the parameters would be: α> 0, α > 0, γ = 0 ad we ca idetify the APE. We ca also say that ω is a exogeous variable i that case. I the secod case: the parameters would be: α> 0, α > 0, γ > 0, we ca t idetify the APE, ω is edogeous ad the bias would be positive. Part B. Properties of estimators Let x, x,..., x be a sample of size from a ormal distributio N ( µ, ). Cosider the followig estimators for µ ad i respect to: biased ubiased, cosistet icosistet. Which of the ubiased are the most ad the least efficiet i fiite samples (have the lowest ad highest variace)? Where do you eed the ormality assumptio? A estimator for a populatio parameter is ubiased if its expectatio equals the populatio parameter, e.g. E ( ˆ µ ) = µ or E( ˆ ) =. I geeral, if ˆΘ is a estimator for Θ, we say that ˆΘ is a ubiased estimator if E( Θ ˆ ) =Θ. Otherwise it is biased. A estimator is cosistet if two coditios hold: ) it is asymptotically ubiased, i.e. p lim( Θ ˆ ) =Θ (or lim( E( Θ ˆ )) =Θ ) ad ) its variace goes to zero as, i.e. lim( V ( Θ ˆ )) = 0 If oe of the coditios does ot hold the estimator is icosistet. We may compare efficiecy of estimators if they estimate the same populatio parameter ad are from the same class, i.e. biased or ubiased. I order to decide which is the most ad which is the least efficiet of the ubiased estimators we have to compare their variaces. Estimators for µ :. ˆ µ = x = xi µ ) = E( x) = E( xi ) = E( xi ) = µ = = µ, because E( x ) = E( x) =... = E( x ) = µ Hece, ˆµ is a ubiased estimator. st year MA i Ecoomics
V ( ˆ µ ) = V ( x) = V ( x ) = V ( x ) = V ( x ) = =, because i i i x, x,..., x are idepedet idetically distributed (i.i.d.) variables cov( x, x,..., x ) = 0, V ( xi ) = V ( xi ) ad V ( x ) V ( x )... V ( x ) = = = =. lim( V ( ˆ µ )) = lim( ) = 0 Hece, ˆµ is a cosistet estimator.,. ˆ µ = xi µ ) = E( xi ) = E( xi ) = µ µ µ + µ µ Hece ˆµ is a biased estimator. The bias is: Bias=B= ) µ = µ = = µ µ lim( E( ˆ µ )) = lim( ) = lim( ) = µ Hece, ˆµ is a asymptotically ubiased estimator. / V ( ˆ µ ) = V ( xi ) = ( ) V xi =, ad ( ) ( ) lim( V ( ˆ µ )) = lim( ) = lim( ) = lim( ) = 0 ( ) ( + ) ( + / ) Hece, ˆµ is cosistet. ˆ. µ = x E( ˆ µ ) = E( x ) = µ ˆµ is ubiased. V ( ˆ µ ) = V ( x ) = lim( E ( µ )) = lim( µ ) = µ ad lim( V ( µ )) = lim( ) = ˆ Hece, ˆµ is icosistet. ˆ ˆ µ = + xi, ω (0,) ω ω ( ) ( ) µ µ ) = E( x+ xi ) = ( E( x ) + E( xi )) = ( µ + ) = ω ω( ) ω ( ) ω ω µ ( ω ) µ ˆµ is biased ad the bias is: B= ) µ = µ = ω ω µ µ lim( )) = lim( ) = ˆµ is asymptotically biased. ω ω ( ) + V ( ˆ µ ) = ( V ( x ) + ( )) ( ) ( ) V xi = + = = ω ( ) ω ( ) ω ( ) ω ( ). x lim( V ( ˆ µ )) = lim( ) = lim( ) = ω ( ) ω ( / ) ω Because ˆµ is asymptotically biased it is icosistet (its variace also does ot go to zero as ). st year MA i Ecoomics
Estimators for :. ˆ = ( xi x) E( ˆ ) = E[( xi x) ] = E[( xi µ ) ( x µ )] = E[( xi µ ) + ( x µ )( x µ ( xi µ ))] Usig the rule: ( a b) = ( a + b( b a)), ad also: E( xi µ ) = ( xi µ ) =, E( x µ ) = ( x µ ), E( xi µ ) = ( xi µ ) = ( xi ) ( µ ) = x µ E( ˆ ) = ( + ( x µ )( x µ ( x µ )) = ( ( x µ ) ) = ( x µ ) But ( x µ ) = V ( x) = ( ) E( ˆ ) = = ˆ is a biased estimator. ( ) lim( E( ˆ )) = lim( ) = ˆ is asymptotically ubiased. The derivatio of the variace of ˆ is o the basis of the distributio of the idempotet o o quadratic form, ( X iµ ) M ( X iµ ), where X is the vector ( x, x,..., x ), M is the idempotet o matrix (equals to its square) M = [ I ii ], ad i is the colum vector (,,,). This form is useful for a sum of squared deviatios. ( ) V ( ˆ ) = V ( ( xi x) ) = ( ) = ( ) lim( V ( ˆ )) = lim( ) = 0 ˆ is cosistet.. ˆ = ( xi x) Usig the results from the previous estimator the expectatio of ˆ is: ( ) E( ˆ ) = E( ( xi x) ) = E( ( xi x) ) = = is ubiased. ˆ i V ( ˆ ) = V ( ( x x) ) =, ad. ˆ = ( x x) E( ) = E( x x) = V ( x x) + [ E( x x)], but ˆ lim( V ( ˆ )) = lim( ) = 0 x+ x+... + x ( ) x E( ˆ ) = V ( x x) = V ( x ) = V ( xi ) = is cosistet. ˆ [ E( x x)] = [ E( x ) E( x )] = [ µ µ ] = 0 ( ) V ( x ) V ( x) + V ( x ) +... + V ( x ) ( ) ( ) ( ) + = + = ˆ is biased. st year MA i Ecoomics
( ) The bias is: B= E( ˆ ) = = ( ) lim( E( ˆ )) = lim( ) = ˆ is asymptotically ubiased. Efficiecy Comparig the variaces of the ubiased estimators for the same populatio parameters, these are ˆµ ad ˆµ oly, we have: V ( ˆ µ ˆ ) = < = V ( µ ), because > ˆµ is the most efficiet estimator of µ ad ˆ is the most efficiet estimator of (because it is the oly ubiased oe). Cosistecy From these results we ca coclude that ot all ubiased estimators are cosistet (e.g. ˆµ is ubiased but icosistet) ad ot all cosistet estimators are ubiased (e.g. ˆ is cosistet but biased). Normality We eed the assumptio for the distributio of X, amely that it is idepedet idetically V ( xi ) = V ( xi ), ad = = = =. We also kow that a liear combiatio of ormal radom distributed (i.i.d.), therefore cov( x, x,..., x ) = 0, V ( x ) V ( x )... V ( x ) variables is ormal. I additio, the Cetral Limit Theorem (CLT) guaratees that if (the umber of observatios) is sufficietly large the samplig distributio of x (ad the distributio of x also) will be approximately ormal. st year MA i Ecoomics 5