Chapter 17: Conditioning information Fan Longzhen Part 3: deep discussion: Estimating and evaluating asset pricing models Discount model is in terms of conditional moments ? The first order condition is ? The expectation is conditional expectation on investor’s time t information; ? The basic pricing equation is ? If the payoff were independent and identically distributed over time, then the conditional expectation is the same as unconditional expectation, we would not worried about the distinction; ? Because the information may be not observable to economist, describing the conditional distribution is not possible, we eventually have to think about unconditional information; ? Unconditional moments are interesting themselves, for example, we may concern why average stock return are high than risk free rate. ])('[)(' 11 ++ = ttttt xcuEcup β )( 11 ++ = tttt xmEp Scaled payoffs ? From the pricing model ? Take unconditional expectations to obtain ? Multiply the payoff and price by any variable or instruments observed at time t, and take expectations ? Sufficiency of adding scaled returns: ? If )( 11 ++ = tttt xmEp )()( 11 ++ = ttt xmEpE t z )()( 11 ttttt zxmEzpE ++ = )()()( 1111 ttttttttttt IxmEpIzzxmEpzE ++++ =?∈?= Conditional and unconditional models in discount factor language ? As an example, consider the CAPM, ? Where is the return on the market or wealth portfolio ? From equation ? Above equation shows explicitly that a and b must vary over time, as ? vary over time; ? If it is to price asset conditionally, the CAPM must be a linear factor model with time-varying weights of the form ? Form the conditional pricing model ? And take unconditional expectation, we have W bRam ?= W R ? ? ? ? ? ? ? ? = += ? ? ? ? ? ? = = + + + + ++ )( )( )( 1 )(1 )(1 1 2 1 1 1 11 W tt f t f t W tt W tt f t f ttt W ttt RR RRE b RbE R a RmE RmE σ f t W tt W tt RRRE ),(),( 1 2 1 ++ σ W tttt Rbam 11 ++ ?= ])[(1 11 ++ ?= t W tttt RRbaE continued ? The conditional model ? only holds if the covariance terms happen to be zero, this is generally not right. ? On the other hand, suppose that it is true that are constant over time, even if are not i.i.d., can be constant over time. Then ? Does implies ),cov(),cov()()()()( ])[(1 111111 11 ++++++ ++ ?+?= += t W ttttt W tttt t W ttt RRbRaRREbEREaE RRbaE ]))()([(1 11 ++ ?= t W ttt RRbEaEE tt ba , W t f t RR , tt ba , ])[(1 11 ++ ?= t W tt RbRaE ])[(1 11 ++ ?= t W t RbRaE Conditional and unconditional in an expected return-beta model ? To put the observation into beta pricing language ? Does not imply ? Even if is a constant, you cannot derive the pricing model also, because t i t f t i tt RRE λβ+= + )( 1 λβγ ii tt RE += + )( 1 λλ = t )](var/),([cov)var(/),cov( 111111 ++++++ ≠= ttt i tttt i t i ffREffRβ A partial solution: scaled factors ? How to handle the parameters change with time ; ? A partial solution is to model the dependence of parameters on variables in the time t information set; ? Let be a vector of variables observed at time t; ? In particular, a linear function of ; ? The discount factor is now as follows: ? ? In place of the one-factor model with time-varying coefficients, we have a three-factor with fixed coefficients. tt ba , tt ba , t z t z )( )( )()( 111010 11010 11 ++ + ++ +++= +++= += tttt ttt tttt fzbfbzaa fzbbzaa fzbzam ),,( 11 ++ tttt fzfz continued ? Since the coefficients are now fixed, we can use the scale-factor model with unconditional moments: ]))([()( ]))([( 1111010 1111010 +++ +++ +++= ?+++= tttttt ttttttt xfzbfbzaaEpE xfzbfbzaaEp