LinearRegression2

9252012 p kolm 4 different functional forms how do we

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Unformatted text preview: © P. KOLM 4 Different Functional Forms How do we incorporate a different functional form? For example: y = b0 + b1 log(x ) + u (assuming x ¹ 0 ) 1 +u (assuming x ¹ 0 ) x log(y ) = b0 + b1x + u (assuming y ¹ 0 ) y = b0 + b1 1 (compared to x + u 1 which would require non-linear regression) Note these parameterizations are linear in x, log(x ) and y = b0 + b1x b2 Many non-linear forms can be given a useful linear representation Interpretation of estimated parameters depends on precise functional form Choice should be based on views of underlying data generation process (statistical tests, convenience) VER. 9/25/2012. © P. KOLM 5 Example: Nonlinear Model of Market Impact Equity market impact model as in Almgren, Thum et al. (2005) Permanent market impact: Find impact parameter g I tperm = g ⋅ st ⋅ nt + etperm Vt (linear regression) Temporary market impact: Find temporary impact parameters (h, b ) I ttemp nt = h ⋅ st ⋅ sign(nt ) ⋅ Vt ⋅ T b + ettemp (non-linear regression) VER. 9/25/2012...
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