Síscs II Degrees Ecoomcs d Mgeme FOMULAE SHEET for STATISTICS II EPECTED VALUE MOMENTS AND PAAMETES - Vr ( E( E( - Cov( E{ ( ( } E( E( E( µ ρ Cov( - E ( b E( be( Vr( b Vr( b Vr( bcov( THEOETICAL DISTIBUTIONS UNIFOM (DISCETE where b coss - Cse... : f ( ( E Vr( m m( m - Cse... m : f ( E ( Vr( m BENOULLI ~ B( θ f ( θ θ ( θ BINOMIAL ~ B( θ ( < θ < E ( θ Vr( θ ( θ f ( θ θ ( θ... ( < θ < E ( θ Vr( θ ( θ I( θ ow θ ( θ Properes: - ~ B( θ ( ~ B( θ - ~ B( θ depede (... ~ B( θ POISSON ~ Po( e f (...! ( > E ( ( Propreres: - Po( Vr I ( ~ depede (... ~ Po( - ~ B( θ wh lrge d smll θ he ~ Po( θ UNIFOM (CONTINUOUS ~ U( f ( < < ( E( Vr( NOMAL ~ N( µ f ( µ ep ( π < < < µ < < < E ( µ Vr( I ( µ ( ow I ( 4 (µ ow Properes: - Sdrd Norml Z ~ N( φ( z φ( z Φ( z Φ( z - ~ N( depede (... ( ~ N µ ~ N µ - ~ N( depede (... ~ N( µ µ µ µ
EPONENTIAL ~ E( ~ E( ~ G( f ( e > > F( e Properes: E ( Vr( I ( - ~ E( depede (... ~ G( d m ~ E( GAMMA ~ G( Gmm fuco: Γ( e d ( > where Γ( ( Γ( > Γ( (! e f ( Γ( Properes: > > E ( Vr( - ~ G( (... depede ~ G( I ( ow - ~ G( c ~ G c > cos c CHI-SQUAE ~ χ ( e f ( > > (eger Γ Properes: - ~ χ ( ~ G - χ ( ~ N( - ~ G( ~ χ ( E ( Vr ( - ~ ( depede (... ~ χ ( χ - ~ N( depede (... ~ ( χ ~ N( ~ χ ( STUDENT - T ~ ( Γ( π U T ~ ( where U ~ N( d V ~ χ ( (depede E ( T Vr ( T ( > V F-SNEDCO ~ F( m U / m F ~ F( m where U ~ χ ( m V ~ χ ( (depede V ( m E ( ( > Vr( ( > 4 m( ( 4 Properes: - ~ F( m ~ F( m - T ~ ( T ~ F( CENTAL LIMIT THEOEM AND COOLLAIES d wh E ( µ d Vr( ~ N( θ Corollry: ~ B( θ depede he ~ N( θ ( θ Couy correco: b θ θ P( b Φ Φ wh eger d b θ ( θ θ ( θ
Corollry: ~ Po( ~ N( whe Couy correco: P ( b b Φ Φ SAMPLING THEO. SAMPLING DISTIBUTIONS SAMPLE MEAN (AVEAGE AND VAIANCE S S S µ E ( Vr( E( S E ( SAMPLING DISTIBUTIONS NOMAL POPULATIONS ~ N( Me Mes dfferece ~ ( Kow vrces ( ( µ ~ N( m Uow bu equl vrces T ( µ ~ ( m ( m ( S m m Vrce S ( ~ χ ( Vrce ro ~ F( m or ~ F( m Z Pred smples ~ ( Z ( - pred smple Z Z LAGE SAMPLES: GENEAL CASE Me ~ N( Mes dfferece ( ( µ ~ m N ( ~ N( / ( ( µ ~ ' ' S S m N ( LAGE SAMPLES: BENOULLI POPULATIONS Proporo Dfferece of proporos Equly of proporos θ ~ N( θ ( θ θ ~ N ( ( ( θ θ ( θ θ ~ N( ~ N( θ( θ θ ( θ ( ( m m ~ N ( where θ( θ m θ m m 3
LAGE SAMPLES: POISSON POPULATIONS Me Mes dfferece Equly of mes ~ N( ~ N( ( ( ~ N( ~ N( m m ~ N( where m m m χ TEST-STATISTIC GOODNESS-OF-FIT TEST m ( N fe Q ~ χ ( m fe po fe Whe prmeers re esmed o ob INDEPENDENCE TEST: r s ( N fe Q ~ χ (( r ( s fe - epeced frequecy for clss he: Q~ χ ( m p o fe NoNo - epeced frequecy for clss LINEA EGESSION MODEL (LM u... y u y... OLS (ordry les sqes esmor Geerl cse Specl cse: y u T T ( y y y... u y y u ( V r( T ( Vr( SST ( Noe: - Vr ( (... y u ( y ( Vr ( from he regresso of u ( o ll remg regressors SST (. 4
Properes:. u (model wh ercep. u (... 3. y u 4. y y (model wh ercep u coeffce: ( ( y y( y y r y y ( y ( y y y where ry y coeffce he model wh ercep: SST SSE SS SST SSE SST SS SST ( y y SSE s he correlo coeffce bewee y d ŷ ( y y SS u SS /( (. SST /( STATISTICAL INFEENCE IN THE LM: y ~ N( u ~ N( ~ ( ( N Vr u ( ~ χ ( ( ~ ( or F ~ F( ( se Specl cses: H : H ~ ( : ~ ( Tes of q ler resrcos o he regresso coeffces H : r Noe: SSr SS F ~ F( q SS q r SS sum of sqed resduls for he resrced model (mposg he q ler resrcos SS sum of sqed resduls for he esrced model. Specl cses Oe resrco (q H r H : θ where θ r r ~ ( Vr ( Tes of zero slopes : F ~ F( θ or ~ ( where θ r θ 5
Tes for o sgfcce of q regressors SSr SS F ~ F( q SS q or r F ~ F( q q Noe: r coeffce for he resrced model coeffce for he esrced model HETEOSKEDASTICIT: T T T T T T Vr( ( Vr( u ( ( ( Whe robus esmor (heerosedscy-cosse vrce esmor T T T V r( ( u (. Iferece o : ~ ( * se ( * where se ( heerosedscy-cosse sdrd error Heerosedscy es: u LM es-ssc: LM ~ χ ( p where d p re respecvely he coeffce d he umber of regressors of he ddol regresso for he es. u PEVISION (FOECAST Averge forecs: E ( y...... θ θ... θ θ ~ ( θ θ Po forecs: y y ~ ( ( se θ y... u y... θ Depede vrble he logrhmc scle - log(y : l og y... - f u ~ N ( y ep( / ep(log y - oher cses y ep(log y where s esmed ddol regresso ESET TEST (FUNCTIONAL FOM SPECIFICATION y... δ y u - es H : δ 3 y... δ y δ y u - es H δ δ : 6