Statistical Analysis: Delta Method
LIMDEP’s WALD command computes values and standard errors for specified linear or nonlinear functions of your estimates. (This is one of the handiest tools in the program.) No need to program derivatives or do any matrix algebra. You provide the command with the (1) name of the vector that contains the parameters, (2) the name of the covariance matrix, (3) the names that you wish to use for the parameters in your function(s) and (4) a set of up to 20 functions of the parameters. The program returns the computed functions, estimated standard errors, asymptotic t ratios and prob values. The name of the command derives from the final computation, which is, if you wish to use it, computation of the Wald statistic for the hypothesis that all of the functions equal zero.
- Use WALD with any model results
- Use with any random vector provided with a covariance matrix
- Up to 20 nonlinear functions
- Use the delta method or the method of Krinsky and Robb to estimate asymptotic covariance matrices
Example: McDonald and Moffitt decomposition
McDonald and Moffitt’s decomposition is an analysis of the conditional mean function in the tobit model. Specifically, in this model, assuming censoring at L in the lower tail only,
It follows that
This breaks the slope into two parts, (1) the change in y given nonlimit times the probability of being above the limit value and (2) the change in the probability of being above the limit times the conditional mean. The formal counterparts to these expressions are:
The following general program illustrates WALD for this computation.
?===================================================================
? Set the model up with these three commands. The rest is standard.
? The routine computes the full set of marginal effects for the
? estimated model, decomposed by the formula given above. Note that
? the message at the end of singular covariance matrix is to be
? expected since the second and seventh functions are equal.
?===================================================================
NAMELIST ; x = the list of Rhs variables $
CREATE ; y = the dependent variable $
CALC ; li = the lower limit value (usually zero) $
?-------------------------------------------------------------------
PROC = McdnM(y,x, li) $
TOBIT ; Lhs = y ; Rhs = x ; Par; Mar $
MATRIX ; xbar = Mean(x) $
CALC ; k = Col(x) $
WALD ; Labels = k_b,st
; Start = b ; Var = varb
; Fn1 = (li - b1'xbar)/st ? Alpha
; Fn2 = 1-Phi(Fn1) ? Prob(noncensored)
; Fn3 = N01(Fn1) / Fn2 ? Lambda
; Fn4 = Fn3*Fn3 - Fn1*Fn3 ? Delta
; Fn5 = Fn2*Fn4 ? First part
; Fn6 = Fn2*(1-Fn4) ? Second part
; Fn7 = Fn5 + Fn6 $ Verify decomposition = Fn2
EXECUTE ; Proc = McDnM(y,x,0)$
+-----------------------------------------------+
| WALD procedure. Estimates and standard errors |
| for nonlinear functions and joint test of |
| nonlinear restrictions. |
| VC matrix for the functions is singular. |
| Standard errors are reported, but the |
| Wald statistic cannot be computed. |
+-----------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
Fncn(1) -1.002830370 .22848675 -4.389 .0000
Fncn(2) .8420286434 .55130622E-01 15.273 .0000
Fncn(3) .2865530305 .84420536E-01 3.394 .0007
Fncn(4) .3694767207 .67567827E-01 5.468 .0000
Fncn(5) .3111099819 .36524564E-01 8.518 .0000
Fncn(6) .5309186615 .91655186E-01 5.793 .0000
Fncn(7) .8420286434 .55130622E-01 15.273 .0000