Non-linear system vs. GMM - same results

[Kraus]

I am estimating a system of equations, once with instruments (GMM with optimal weighting matrix), once without them (non-linear system with a diagonal weighting matrix). However, I get precisely the same results. Here's the code

Code: Select all

INSTR CONSTANT VIX{1} US_SLOPE{1} EU_SLOPE{1} CN_SLOPE{1} UK_SLOPE{1} SW_SLOPE{1} JP_SLOPE{1}
NONLIN(PARMSET=P) C1 C2 C3 C4 C5 B11 B71 B51 B21 B31 B12 B72 B52 B22 B32

FRML F11 US_10 = C1 + B11*US_SLOPE{1} + B12*VIX{1}
FRML F12 EU_10 = C1 + B11*EU_SLOPE{1} + B12*VIX{1}
FRML F13 CN_10 = C1 + B11*CN_SLOPE{1} + B12*VIX{1}
FRML F14 UK_10 = C1 + B11*UK_SLOPE{1} + B12*VIX{1}
FRML F15 SW_10 = C1 + B11*SW_SLOPE{1} + B12*VIX{1}
FRML F16 JP_10 = C1 + B11*JP_SLOPE{1} + B12*VIX{1}

FRML F71 US_7Y = C2 + B71*US_SLOPE{1} + B72*VIX{1}
FRML F72 EU_7Y = C2 + B71*EU_SLOPE{1} + B72*VIX{1}
FRML F73 CN_7Y = C2 + B71*CN_SLOPE{1} + B72*VIX{1}
FRML F74 UK_7Y = C2 + B71*UK_SLOPE{1} + B72*VIX{1}
FRML F75 SW_7Y = C2 + B71*SW_SLOPE{1} + B72*VIX{1}
FRML F76 JP_7Y = C2 + B71*JP_SLOPE{1} + B72*VIX{1}

FRML F51 US_5Y = C3 + B51*US_SLOPE{1} + B52*VIX{1}
FRML F52 EU_5Y = C3 + B51*EU_SLOPE{1} + B52*VIX{1}
FRML F53 CN_5Y = C3 + B51*CN_SLOPE{1} + B52*VIX{1}
FRML F54 UK_5Y = C3 + B51*UK_SLOPE{1} + B52*VIX{1}
FRML F55 SW_5Y = C3 + B51*SW_SLOPE{1} + B52*VIX{1}
FRML F56 JP_5Y = C3 + B51*JP_SLOPE{1} + B52*VIX{1}

FRML F21 US_2Y = C4 + B21*US_SLOPE{1} + B22*VIX{1}
FRML F22 EU_2Y = C4 + B21*EU_SLOPE{1} + B22*VIX{1}
FRML F23 CN_2Y = C4 + B21*CN_SLOPE{1} + B22*VIX{1}
FRML F24 UK_2Y = C4 + B21*UK_SLOPE{1} + B22*VIX{1}
FRML F25 SW_2Y = C4 + B21*SW_SLOPE{1} + B22*VIX{1}
FRML F26 JP_2Y = C4 + B21*JP_SLOPE{1} + B22*VIX{1}

FRML F31 US_3M = C5 + B31*US_SLOPE{1} + B32*VIX{1}
FRML F32 EU_3M = C5 + B31*EU_SLOPE{1} + B32*VIX{1}
FRML F33 CN_3M = C5 + B31*CN_SLOPE{1} + B32*VIX{1}
FRML F34 UK_3M = C5 + B31*UK_SLOPE{1} + B32*VIX{1}
FRML F35 SW_3M = C5 + B31*SW_SLOPE{1} + B32*VIX{1}
FRML F36 JP_3M = C5 + B31*JP_SLOPE{1} + B32*VIX{1}

NLSYSTEM(INSTR,PARMSET=P,SWOUT=WGHTMTRX) 1980:1 2014:5 F11 F12 F13 F14 F15 F16 $
F71 F72 F73 F74 F75 F76 F51 F52 F53 F54 F55 F56 F21 F22 F23 F24 F25 F26 $
F31 F32 F33 F34 F35 F36

And then I do the same estimation again without the INSTR option. When looking at the SWOUT-matrix, it is definitely not diagonal. So is there an opimized weighting in both cases? What do I get wrong here? Thanks for your help.

[TomDoan]

NLSYSTEM(INSTR) without other options is doing a factored weight matrix. In this case, that gives exactly the same result as NLSYSTEM without INSTR because the orthogonality conditions for non-linear minimum distance are in this case, exactly the same as the ones for your instruments. (Your instrument list is exactly the list of explanatory variables across the system). If you add the ZUDEP option, you'll get different estimates because the weight matrix doesn't factor.