Indirect Inference (II)

Indirect Inference(II) provides a classical statistical inferential framework for testing a model. The aim is to compare the performance of the auxiliary model estimated on the simulated data derived from the model, with the performance of the auxiliary model when estimated from the actual data. In practice we use a VAR as the auxiliary model, but you could also use IRFs and moments. The criterion we use when evaluating the model is the Wald test of the differences between the vector of relevant VAR coefficients from simulated and actual data. If the DSGE model is correct then it should produce simulated data that is similar to the actual data, and therefore the VAR estimates on the simulated data will not be significantly different from the VAR estimates on the actual data.

From the actual data we get the VAR coefficients , and from the simulations we get N sets of VAR coefficients (for i = 1:N), from which we perform the relevant calculations. The Wald statistic that we calculate is: where and Programme

Matlab programme package INDIRECT to implement this method now available for download. The method is carefully explained for users in Le et al. (2016) which should be cited when using any of these programmes.

Unzip the file, you will find a user's manual, a readme file, three main matlab .m files (II_Wald_Test.m,  II_Wald_Power.m and II_Wald_Estimation.m) and a few built-in functions.

1. Model Evaluation by II(I_Wald_Test.m)

The following steps summarise how to implement the II test by bootstrapping. Each step is programmed by a function.

Step 1: Calculate the residuals and innovations of the economic model conditional on the data and parameters.

{inno, res, rho_est}=Get_Res_LIML()

Step 2: Derive the simulated data by bootstrapping(simulating with model's own implied shocks, solving by DYNARE)

{boots_data}=Boots_data()

Step 3: Compute the Wald statistic.

{pvalue, Wald, Trans_Wald}=Wald_stationary()

2. Power of II test(II_Wald_Power.m)

We examine the power of the II wald test by positing a variety of false models, increasing in their order of falseness. We generate the falseness by introducing a rising degree of numerical mis-specification for the model parameters. Thus we construct a False DSGE model whose parameters were moved x% away from their true values in both directions in an alternating manner (even-numbered parameters positive, odd ones negative); similarly, we alter the higher moments of the error processes (standard deviation) by the same +/ − x%. We may think of this False Model as having been proposed as potentially ‘true’ following previous calibration or estimation of the original model. The transformed Wald is calculated each time. The power of the test is the probability of rejecting a false model by the data (or the probability that Transformed Wald is bigger than 1.645).

s=1:1000
{pvalue(s), Wald(s), Trans_Wald(s)}=II_Wald_stationary()
power=mean(Trans_Wald>1.645)

3. Model Estimation by II(II_Wald_Estimation.m)

The Wald statistic measures the distance between the data and the model. Therefore to estimate the model parameters by II, we can use any minimising algorithm to minimise the Wald for the actual data. The function to minimise takes the coefficients as an input and then does Steps 1–3 above, giving the Wald as the output. The fminsearchbnd algorithm supplied in Matlab is suggested, as it has been found to find global minima.

{Trans_Wald } = CalcWald()
{II_coef} =fminsearchbnd(@(II_coef)  CalcWald())

For the details of how to use this programme, see II test user manual.

There are also 2 examples attached in INDIRECT programmes. One is Smets and  Wouters (2007) NK model. Another is 3-equation NK model. Both of are written in the dynare .mod files.

Literature

MMF(2015) special Lecture.

MMF(2015) special Lecture slides.

Le, V.P.M., Meenagh, D., Minford, P., Wickens, M., and Xu,  Y, 2016. Testing macro models by indirect inference: a survey for usersOpen Economies Review 27,1-38

Le, V.P.M., Meenagh, D., Minford, P., Wickens, M., 2011. How much nominal rigidity is there in the US economy - testing a New Keynesian model using indirect inference. Journal of Economic Dynamics and Control 35(12), 2078-2104.

Meenagh, D., Minford, P.,Wickens, M. R., and Xu, Y, 2015. Comparing Indirect inference and likelihood testing methods: asymptotic and small sample results. Working paper, Cardiff Economics Working Papers No E2015/4, Cardiff University, Cardiff Business School, Economics Section.