Ec304 derivation of the t-statistic distribution PDF

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Ec304.'Derivation'of'the't-statistics.' ' Interval'estimation'and'hypothesis'testing'involve'the'Student't-distribution.' ' Why'does'the't-statistic'that'arises'in'testing'hypotheses'about'the'true'value'of'the' slope'coefficient'in'the'simple'regression'model'have'a'Student't'distribution'with' (n–2)'degrees'of'freedom?'' Note,&in&multiple®ression&model&the°rees&of&freedom&become&(n–k),&where&n&is&the& sample&size&and&k&is&the&number&of&the®ression¶meters,&the&intercept&and&slope& coefficients.&Here,&I&am&referring&to&a&simple®ression&case,&with&one&slope&coefficient.& ' The'answer'hinges'in'the'procedure'of'the'derivation'of'the't-statistic'in'testing'of' hypotheses'about'the'slope.'' ' Assumptions'for'the'proof:'assume'a'simple'regression'model'with'one'explanatory' variable'and'an'intercept;'assume'that'the'sample'size'is'large'and'the'error'term'is' normally'distributed'with'a'constant'(unobserved)'variance.'' ' The'slope'coefficient'is'assumed'to'be'normally'distributed'if'the'sample'is'large' (≈n≥100)'(CLT):'' a1#~#N(β,#σu2/Σ(xi#–#x)2).'A'standardized'version'is'obtained'by'subtracting'its'mean' and'dividing'by'its'standard'deviation:''' ' Z'='{(â1'–'a1)/√[#σu2/#Σ(xi#–#x)2]}'~#N(0,#1).## # Since'we'usually'do'not'know'the'true'error'variance'(σu2),'we'use'its'sampling' counterpart'(su2):' t'='{(â1'–'a1)/√[#s2/#Σ(xi#–#x)2]}.'This'is'so-called't-statistic.'' ' It'has't'distribution'with'(n–2)'d.f.'that'can'be'approximated'by'the'normal' distribution'if'the'sample'size'is'large'enough.'(n–2)'arises'from'the'fact'that'when' the'error'term'is'normally'distributed,'ui''~'N(0,'σu2),'its'standardized'version'is' given'by'(ui/σ)~'N(0,'1).' ' The'square'of'a'standard'normal'variable'is'a'chi-square'random'variable'with'one' degrees'of'freedom''(d.f.),'so'(ui/σ)'2~'χ21'(by'the'definition'of'the'random'variable' that'has'a'χ2'distribution).'' ' If'all'the'errors'are'independent,'then'it'has'n'degrees'of'freedom:'Σ(ui/σ)'2~'χ2n.'' ' But'because'the'true'random'errors'are'unobservable,'we'replace'them'by'their' sample'counterparts,'the'least'squares'residuals'ui'='Yi'–'a'–'bAi'.' ' It'yields'Σui2/σu2'='(n–2)su2/σu2'(from'the'formula'for'the'square'of'the'standard' error'of'the'regression:'su2#='Σui2/(n–2)).'' '

The'expression'Σui2/σu2'='(n–2)su2/σu2'does'not'have'a'χ2n'distribution'because'only' (n–2)'residuals'are'independent.'' ' Therefore,'when'multiplied'by'(n–2)/σu2,'the'random'variable'su2'has'a'χ2n–2' distribution:'' (n–2)su2/σu2'~'χ2n'–2.' ' This'χ2n'–2'random'variable'and'Z'='{(#â1'–'a1)/√[#σε2/Σ(Ai#–#Ā)2]}'~#N(0,#1)#'are' independent'(according'to'the'assumption'that'the'sum'of'regression'residuals'is' independent'of'the'distributions'of'the'regression'coefficients)'and'therefore'their' ratio'has'a't-distribution'with'(n–2)'d.f.:'' ' tn–2'='Z/√['χ2n'–2'/(n–2)]' ' Rearranging'this'ratio,'we'obtain'the'familiar'expression'for'the't-statistic'for' testing'statistical'hypotheses:' ' tn–2'='Z/√[V/(n–2)]'='(â1'–'a1)/√(σu2/#Σ(xi#–#x)2)/[√{(n–2)su2/σu2/(n–2)}]'='' ='(â1'–'a1)/√(su2/Σ(xi#–#x)2).'=''(â1'–'a1)/(SE(â1)'~'tn–2.'' ' Q.E.D.' '...