*SHAZAM PROGRAM FILE TO TO SIMULATE FORWARD AND SPOT EXCHANGE RATE
* TIME PATHS UNDER DIFFERENT ASSUMPTIONS ABOUT AGENT'S INFORMATION
*
* The program assumes that the real exchange rate is a random walk and
* that the domestic and foreign inflation rates exhibit zero trend.
*
* [NOTE: THESE COMMENTS WILL NOT ALWAYS CORRECTLY APPEAR IN THE SHAZAM
*  OUTPUT.  IT SEEMS THAT SHAZAM SWALLOWS SOME OF THE MATHEMATICAL
*  NOTATION IN A MISGUIDED ATTEMPT TO EXECUTE IT.]
*
* The innovation in the time-series process generating the real exchange
* rate is assumed to be composed of two parts---a random component that
* agents can know about with error, and a random component that is pure
* news.  We thus generate two random terms E1 and E2 and assume that
* agents' basic exchange rate prediction is a weighted average of these
* to error terms with G being the weight attached to E1 and (1-G) being
* the weight attached to E2.  Larger values of G imply that the proportion
* of future exchange rate innovations that agents can forecast is
* correspondingly larger.  In order to ensure that the variance of the
* shock to the real exchange rate is the same whatever agents know 
* about it the G and (1-G) weights have to be scaled to ensure that their
* squared values sum to unity.  This is done by dividing both raw weights
* by a scale factor S equal to the square root of the sum of the square of 
* G and the square of (1-G).  That is, the scale factor will equal 
*
*       S = sqrt ( [ G^2  + (1-G)^2] )    [^ means "to the power of"]
*
* so that the scaled weights will be G/S and (1-G)/S.  This scaling follows
* from the fact that the variance of the weighted sum of two independent 
* random variables
*
*       T = [G/S] E1(t) + [(1-G)/S] E2(t)
*
* equals
*
*     Var{T} = [G/S]^2 Var{E1} + [(1-G)/S]^2 Var{E2}.
*
*
* For Var{T} to always be unity, given that Var{E1} and Var{E2} are unity
* by construction, it is required that 
*
*     (G^2)/(S^2) + [(1-G)^2]/(S^2) = ((G^2) + [(1-G)^2])/(S^2) = 1.
*
* This is guaranteed by our definition of S.
*
* In addition to the two random variables E1(t) and E2(t), we need a third
* random variable U(t) to incorporate agents' prediction errors.  These
* errors, also having unit variance by construction, add to agents forward
* rate prediction based on the weighting of the error terms E1(t) and E2(t).
* The variance of U(t) is scaled by the factor Z to incorporate the degree
* of prediction error desired in any given simulation---Z can range from
* zero upward.
*
* FIRST WE SET THE SAMPLE SPACE 
*
SAMPLE 1 300
*
* EDIT THE TWO LINES BELOW TO IMPOSE ON THE SIMULATION RUN THE DESIRED
* VALUES OF G AND Z.  THE CLOSER G IS TO 1.0 THE GREATER THE FRACTION
* OF THE EXCHANGE RATE INNOVATION POTENTIALLY PREDICTABLE BY AGENTS.
* THE BIGGER Z IS ABOVE 0.0, THE BIGGER AGENTS' PREDICTION ERRORS---A
* VALUE OF Z = 1.0 IMPOSES A VARIANCE OF AGENTS' PREDICTIONS OF UNITY,
* WHICH IS THE SAME AS THE VARIANCE OF THE INNOVATIONS TO THE REAL 
* EXCHANGE RATE.
GEN1 G = 0.0
GEN1 Z = 0.5
*
GEN1 S = (sqrt((G**2)+((1-G)**2)))
GEN1 CHKNUM = (G**2)/(S**2)+((1-G)**2)/(S**2)
GEN1 CHKNUM
*
* When we give the regression coefficients in the forward discount
* hypothesis an errors in the variables interpretation, we can show that
* the true value of the coefficient b1 equals the ratio of [(G^2)/S^2)] to
* ( [(G^2)/(S^2)] + Z^2 ).  Here Z^2 is the variance of the prediction 
* error (which equals the square of the scale factor Z times the variance
* of U(t) which is unity by construction.
* 
* SO WE CALCULATE THE TRUE VALUE OF THE COEFFICIENT b1 as
*
GEN1 B1 = (G**2/S**2)/((G**2/S**2)+Z**2)
*
* NEXT WE CONSTRUCT THE THREE ERROR TERMS
*
GENR E1 = NOR(1)
GENR E2 = NOR(1)
GENR U = NOR(1)
*
* NEXT WE CONSTRUCT THE REAL EXCHANGE RATE SERIES
* 
GENR REX = (G/S)*E1+((1-G)/S)*E2
SAMPLE 2 300
GENR REX = LAG(REX)+(G/S)*E1+((1-G)/S)*E2
*
* AND THEN THE FORWARD EXCHANGE RATE SERIES
*
GENR FPREX = LAG(REX)+(G/S)*E1+Z*U
*
* THE CHANGE IN THE SPOT RATE FROM t-1 TO t IS
*
GENR CHGREX = REX-LAG(REX)
*
* AND THE LAGGED FORWARD DISCOUNT OR FORWARD PREDICTION OF THE CHANGE
* IN THE SPOT RATE IS
*
GENR LAGFDISC = FPREX-REX
*
* NOW WE PLOT THE SERIES. 
*
* COMMENT OUT THE THREE LINES BELOW TO SUPRESS THE PLOTS
PLOT REX FPREX / TIME
PLOT REX FPREX
PLOT CHGREX LAGFDISC
*
* RUN THE REGRESSION POSED BY THE FORWARD RATE HYPOTHESIS
*
OLS REX FPREX / NOCONSTANT COEF=ESTBO STDERR=SEBO
*
* EXTRACT AND RENAME IMPORTANT STATISTICS
* THE R2
GEN1 RSQFR=$R2
* THE MEAN SQUARED ERROR
GEN1 MSE=$SSE/$DF 
* THE ESTIMATE OF B0 IS GIVEN BY THE VARIABLE ESTBO
* THE STANDARD ERROR OF ESTIMATED B0 IS GIVEN BY THE VARIABLE SEBO  
* THE t-ratio FOR THE DEVIATION OF b0 FROM UNITY -- FORWARD RATE HYPOTHESIS
GEN1 TRATFR = (1-ESTBO)/SEBO
*
* RUN THE REGRESSION POSED BY THE FORWARD DISCOUNT HYPOTHESIS
*
OLS CHGREX FPREX / NOCONSTANT COEF=ESTB1 STDERR=SEB1 
*
* EXTRACT AND RENAME IMPORTANT STATISTICS
* THE R2
GEN1 RSQFD=$R2
* THE MEAN SQUARED ERROR
GEN1 MSE=$SSE/$DF
* THE ESTIMATE OF B1 IS GIVEN BY THE VARIABLE ESTB1
* THE STANDARD ERROR OF ESTIMATED B1 IS GIVEN BY THE VARIABLE SEB1
* THE t-ratio FOR THE DEVIATION OF ESTIMATED B1 FROM TRUE B1 --- FORWARD
*    DISCOUNT HYPOTHESIS (ERRORS IN VARIABLES INTERPRETATION)
GEN1 TRATFD = (ESTB1 - B1)/SEB1  
*
* PRESENT CALCULATED STATISTICS
*
GEN1 ESTBO 
GEN1 TRATFR 
GEN1 RSQFR
*
GEN1 B1 
GEN1 ESTB1 
GEN1 TRATFD 
GEN1 RSQFD
*
* COMMENT OUT THE TWO LINES BELOW TO PREVENT THE DATA GENERATED IN THIS
* SIMULATION RUN FROM BEING PRINTED
WRITE(REXSIM.MAT) REX FPREX CHGREX LAGFDISC
*
STOP