I'm interested in hearing your experience on these two estimation techniques specially on the choice of moments and auxiliary statistics.
METHOD OF SIMULATED MOMENTS: MSM
MSM is an estimation method based on simulation. MSM allows only the comparison of moments between the observed response and expected response. The classical approach by McFadden is given by:
[1] T = (Instrument vector) X (observed response) - (estimated response)
… where T = theta for estimated moment. Formally, MSM may be written as:
[2] g(T) = m^T – ms(T)
… where g(T) = estimated moment at T ;m^T = vector simulated moment at T; and ms(T) = expected moment at T
The expected T or T^ is given by:
[3] T^ = arg min Q(T)
See Daniel McFadden (1989). "A method of simulated moments for estimation of discrete response models without numerical integration." Econometrika, vol. 57, No. 5 (Sept 1989), pp. 995-1026.
INDIRECT INFERENCE
It is also an estimation method based on simulation and it is equivalent to the maximum likelihood probability function. The estimation is based on simulation; indirect inference attempts to estimate parameters of economic models through the aid of auxiliary model. An auxiliary model is a model that gives an approximation of the function would best describe the data set. The result produced by the auxiliary model and the estimate derived from the simulation must be as close as possible.
Assume that the economic model is given by:
[4] Y = G(x, u, beta)
… where x = exogeneous data, y = endogeneous data, u = random error, and beta = parameter vector with k-dimension.
The parameter of the auxiliary function is given by:
[5] T^ = arg max(T) SUM log f(y | y’, x, T)
DIFFERENCE BETWEEN MSM AND INDIRECT INFERENCE
Firstly, MSM allows only the estimation of moment, i.e. mean or average of data. Indirect inference, on the other hand, allows the use of any aspects of the data to compare moments between the observed and estimated moments. For example, the student t-equation is a good example of MSM because the basis of the t-equation is:
[6] t = (x^ - mu) /S /sqrt n
… where x^ = observed mean; mu = assume population mean (expected mean in McFadden); S = observed standard deviation and n = sample size. The moment (or mean mu) may be estimated thus:
[7] mu = t(S/sqrt n) – x^
The unit normal distribution Z-equation would be a good illustration of indirect inference. The Z-equation is given by:
[8] Z = (x^ - mu) / sigma / sqrt n
… where x^ = observed mean; mu = estimated mean; sigma = estimated standard deviation and n = sample size.
Secondly, another difference between the MSM and indirect method is the objective of the argument. MSM’s argument is to MINIMIZE THE DIFFERENCE between the observed and expected estimation of the moment. On the other hand, the objective of indirect inference is to MAXIMIZE THE LIKELIHOOD that the estimate moment value between the observed vector and expected value under the auxiliary model be as possible.
METHOD OF SIMULATED MOMENTS: MSM
MSM is an estimation method based on simulation. MSM allows only the comparison of moments between the observed response and expected response. The classical approach by McFadden is given by:
[1] T = (Instrument vector) X (observed response) - (estimated response)
… where T = theta for estimated moment. Formally, MSM may be written as:
[2] g(T) = m^T – ms(T)
… where g(T) = estimated moment at T ;m^T = vector simulated moment at T; and ms(T) = expected moment at T
The expected T or T^ is given by:
[3] T^ = arg min Q(T)
See Daniel McFadden (1989). "A method of simulated moments for estimation of discrete response models without numerical integration." Econometrika, vol. 57, No. 5 (Sept 1989), pp. 995-1026.
INDIRECT INFERENCE
It is also an estimation method based on simulation and it is equivalent to the maximum likelihood probability function. The estimation is based on simulation; indirect inference attempts to estimate parameters of economic models through the aid of auxiliary model. An auxiliary model is a model that gives an approximation of the function would best describe the data set. The result produced by the auxiliary model and the estimate derived from the simulation must be as close as possible.
Assume that the economic model is given by:
[4] Y = G(x, u, beta)
… where x = exogeneous data, y = endogeneous data, u = random error, and beta = parameter vector with k-dimension.
The parameter of the auxiliary function is given by:
[5] T^ = arg max(T) SUM log f(y | y’, x, T)
DIFFERENCE BETWEEN MSM AND INDIRECT INFERENCE
Firstly, MSM allows only the estimation of moment, i.e. mean or average of data. Indirect inference, on the other hand, allows the use of any aspects of the data to compare moments between the observed and estimated moments. For example, the student t-equation is a good example of MSM because the basis of the t-equation is:
[6] t = (x^ - mu) /S /sqrt n
… where x^ = observed mean; mu = assume population mean (expected mean in McFadden); S = observed standard deviation and n = sample size. The moment (or mean mu) may be estimated thus:
[7] mu = t(S/sqrt n) – x^
The unit normal distribution Z-equation would be a good illustration of indirect inference. The Z-equation is given by:
[8] Z = (x^ - mu) / sigma / sqrt n
… where x^ = observed mean; mu = estimated mean; sigma = estimated standard deviation and n = sample size.
Secondly, another difference between the MSM and indirect method is the objective of the argument. MSM’s argument is to MINIMIZE THE DIFFERENCE between the observed and expected estimation of the moment. On the other hand, the objective of indirect inference is to MAXIMIZE THE LIKELIHOOD that the estimate moment value between the observed vector and expected value under the auxiliary model be as possible.
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