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About the article
Published Online: 2013-09-26
Published in Print: 2014-04-01
It would be estimated sequentially, using the residuals of previous steps as instruments for the endogenous regressors. Specifically, as an instrument for in the price equation, as an instrument for in the net tax equation, and as an instrument for in output equation.
In Appendix C be we show that the estimated structural shocks resulting from (4) and (4′) fully coincide for net taxes and expenditure, and coincide except for a scale factor for output and prices.
Primiceri (2005) suggests a more general procedure in case several factorizations, i.e., orderings of the variables, appear plausible. This is to impose a prior on each of them, and then to average the results obtained on the basis of posterior probabilities.
Except for the initial state of log dt whose covariance matrix is set to a multiple of the identity.
The corresponding value for Qd was set to (0.01)2 and the ones for to (0.1)2, following Primiceri (2005).
This is implemented in such a way that the whole history of θ’s generated at step 1 is discarded, in case the condition is not met at least for one t.
For data sources and the precise way fiscal variables are computed, see Appendix A.
See “Effect of the ARRA on Selected Federal Government Sector Transactions,” on the BEA website at http://www.bea.gov/recovery/index.htm?tabContainerMain=1.
We follow the usual practice of presenting a simplified version of impulse-responses, in which the response for shocks at t is a function of the parameters estimated for that date all steps ahead.
Not shown but available from the authors on request.
Note that the estimates depicted in Figure 3 refer to the second quarter of each year, and the trough of the 1973–1975 recession was in the first quarter.
The reason may be that, although Kirchner, Cimadomo, and Hauptmeier (2010) do not impose the stability condition, they use a smoothed variant of the simulation procedure. We use a filtered variant instead.
These are computed as follows. A time-invariant reduced form VAR is estimated for each of the rolling-samples. On the basis of the point estimate for the covariance matrix, one draws firstly for this matrix, assuming a inverse-Wishart distribution. The structural decomposition is applied to each draw. At the same time, one draws for the vector of coefficients, assuming a Gaussian distribution, conditional on the covariance matrix previously drawn. The implied impulse-responses are obtained on the basis of 1000 draws and the relevant statistics computed.
It is hard to blame the size of the rolling window (25 years) for this instability. For instance, although in a simpler context, Stock and Watson (2007) use rolling samples with only 10 years. The uncertainty surrounding the point estimates in Figure 4 is not unusually large for VAR standards.
The size of the multipliers in these models depends, for instance, on the intensity of the (negative) relationship between the markup ratio and output and the (positive) elasticity of labor supply (Hall 2009), or the proportion of non-Ricardian consumers (Galí, Lopez-Salido, and Vallés 2007).
It is worth noting that the size of output (and price) shocks in the identification scheme (4′), which we use in the simulations, does not coincide with the one in (4); see the Appendix B on this. However, since this difference is small — the standard deviation of the shocks is about 4% bigger in the first scheme in a fixed-parameter setting — we ignore this issue.
This behavior is explained as follows. In the course of recessions there is a large decrease in net taxes, which results from the simultaneous fall in taxes and rise in social benefits. Therefore, the weight of taxes in total goes up and that of transfers, which is negative, becomes more negative. Since the elasticity of taxes to output is positive and the elasticity of transfers is negative, by itself this leads to an increase in the overall elasticity.
This is obtained as the difference between the products of the response of each fiscal variable and the ratio of that variable to GDP. Note that our semi-elasticity actually refers to the primary deficit, since the definition of fiscal variables we adopt excludes interest outlays.
Consumption of fixed capital is excluded on two grounds. Firstly, there are no shocks to this variable, which is fully determined by the existing capital stock and depreciation rules. Secondly, from the viewpoint of the impact on aggregate demand, it is the cost of capital goods at time of acquisition (already recorded in government investment) that matters and not at time of consumption.
This description of the simulation procedure assumes that the covariance matrix of the state innovations, Qd, is unrestricted and thus the volatility states are drawn jointly. One could alternatively assume a diagonal Qd matrix – i.e., independent state innovations –, in which case the simulations would be carried out equation by equation. We experimented with both possibilities and the results were similar.
In the five-variable system including private consumption, T0 is set to 56. This is equal to the size of the vector θt plus 1, the minimum number of degrees of freedom for the prior to be proper (and exceeds the number of observations in the training sample).