Data Generating Processes verification

It is essential to confirm that the common underlying structure of DGPs generates the needed information for the various risk-sharing estimation approaches. More precisely, we must confirm whether the three data formats used in the empirical estimation are supported by the DGPs: first differences, levels, and SVAR.

For the risk-sharing estimation in first differences, since output and consumption are generated in levels, we need to verify that the conversion from level into first difference does not alter the outcome of the risk-sharing estimation by showing that the estimation equation in first difference in the MCS is the same as Eq. (2.18) and (4.2). To do so, we start with taking first difference of Eq. (5.5), which gives us

∆c = ∆yβu+ ∆D (5.16)

error in Eqs. (2.18) and (4.2), υ, that could contain various factors that directly influence consumption. Essentially, in Eq. (5.16) the general error which is υ in Eqs. (2.18) and (4.2) is limited to demand shocks, D_{. Nonetheless, estimating}

Eq. (5.16) using OLS would yield the same consistent unshared risk estimate of βu, the variance covariance of output and consumption, if there is no endogeneity,

and an inconsistent estimate if endogeneity is present, as when estimating Eqs. (2.18) and (4.2) using OLS. In addition, the common structure of DGPs also allows for endogeneity bias in first difference. To see this take Eq. (5.2) and apply first difference such that we get

∆y = y(α − 1) + (Dψ + S) (5.17)

Given that D _{is still present in Eq. (5.17), output is still not independent of the}

error, D, in Eq. (5.16) such that E(yD) 6= 0.

In regards to the Level-method, we need to verify that in Eq. (5.5) both c and y, even after having being demeaned, have the potential to be non-stationary. To verify that unit root is possible we start with the output Eq. (5.2):

yt = yt−1α + y (5.18)

where, for simplicity, demand and supply shocks are combined in y such that y = ψD+ S. The equivalent of Eq. (5.18) for the mean output is given by

¯

yt= wyt= w(yt−1α + y) (5.19)

such that if we demean Eq. (5.18) by subtracting Eq. (5.19) we get

yt− ¯yt = (yt−1− ¯yt−1)α + y− ¯y (5.20)

so that if |α| = 1, the question becomes how (yt− ¯yt−1) behaves. In other words,

the question is whether the demeaned consumption and output term for individual countries converges to zero or whether it diverges from zero.

The answer to the question lies in the weighting process. Let us start, for sim- plicity, by assuming that the weights are identical for all countries: wi = wj. Also,

given that y = ψD + S and both D and S are i.i.d. N (0,1) then it follows that y ∼ i.i.d. N (0, 1). So that when taking the average across the countries, the weighted mean tends to converge, as the sample of countries is increased, to the cho- sen mean of the symmetrically distributed errors, the supply and demand shocks, which is zero, i.e. lim

N →∞ N

−1PN i=1

y _{= E(}y_{) = 0. This implies that the aggregate}

of output shocks is close to zero in each period. Thus ¯yt grows potentially more

shock tends more towards zero compared to country individual shocks, and thereby providing potential for unit-root in the demeaned variable for some of the 24 coun- tries. On the other hand, the weights in the DGPs are not identical for each country and the panel dimension is limited to 24, which means that the weighted mean of the shocks does not have to be equal or converge to the mean of the symmetric distribution from which the supply and demand shocks are randomly chosen, but rather be close to the shocks experienced by the largest countries.7 _{Nonetheless unit}

roots can exist for some countries as the common structure of the DGPs does not prohibit some countries growing faster or more slowly than the mean growth rate. This implies that if we ran country individual level estimation, the possibility exists that some countries would have unit roots and thus we could get consistent country individual level estimates, while for other countries there wouldn’t be a unit root and thus least square estimation using the levels of variables would provide inconsistent risk-sharing parameter estimates. Thus the panel Level estimate can, but does not have to, give consistent estimates when |α| = 1. Alternatively, if |α| 6= 1, then there is no potential for a unit root, whether in ratio or not.

Last but not least, we need to verify that output allows for the SVAR to identify supply and and demand shocks. To see whether the SVAR identifies the shocks, we need to ensure that the the contemporaneous impact of supply and demand shocks can be appropriately identified, subject to the long run restriction and the impact of lagged endogenous variables. Thus we need to verify:8

C = (I − A1− A2)−1A−10 B

where A1 and A2 are the (2 × 2) coefficient matrices of the lagged endogenous

variables, A0 is the (2 × 2) matrix of contemporaneous effect of the endogenous

variables (which we assumed is a I-matrix in the empirical application), and B is the contemporaneous impact of supply and demand shock on output and price matrix (2 × 2). Even though we know that all countries will be an AR(1) process (given the DGPs), the reason that we have an AR(2), is that an AR(2) process has been uniformly applied to capture the serial correlation for all countries in the actual empirical application in the previous chapter. Therefore, the application of an AR(2) process reflects the empirical application, and it is in this spirit that the MCS applies an AR(2) condition. Rewritten, we get9

(I − A1− A2)C = B

7_{This does not mean that as N, the sample of countries, increases and the weight of each country}

drops that the mean would not converge to zero, although in our case N is not large enough for that to hold.

8_{A good overview and derivation of the long run restriction matrix can be found in L¨utkepohle}

(2005).

9_{As A}−1

0 = I, it follows that A −1

when writing out the matrices " 1 0 0 1 # − " a1₁₁ a1₁₂ a1 21 a122 # − " a2₁₁ a2₁₂ a2 21 a222 # " c11 c12 c21 c22 # = B

where, given the long-run restriction of demand shocks having no impact on output, c11= 0, we get " 1 0 0 1 # − " a1₁₁ a1₁₂ a1₂₁ a1₂₂ # − " a2₁₁ a2₁₂ a2₂₁ a2₂₂ # " 0 c12 c21 c22 # = B

which implies that

B = "

c21(−a112− a212) c12(1 − a111− a211) + c22(−a112− a212)

c21(1 − a122− a222) c12(−a211 − a221) + c22(1 − a122− a222)

#

where b11, the contemporaneous effect of demand shocks on output, is equal to

c21(−a112− a212).

Effectively, the contemporaneous effect of demand shock on output depends on the coefficient of price in the output estimation and the long run impact of demand shock on price. However, as output was defined independently of price, the coeffi- cients a1₁₂ and a2₁₂ will be equal to zero, and so that b11 = 0.10 This implies that

demand shocks do not affect output contemporaneously, or rather that that demand shocks cannot be separately identified from supply shocks. So, given the common structure of the DGPs process, the SVAR would not be able to properly identify supply and demand shocks due to the lack of interaction of output and price in the DGPs.

This leaves two options: re-define the DGPs or use a B-model SVAR. The first option entails making the common structure of the DGPs more dynamic by introduc- ing greater interaction between the price and output, rendering the interpretation of the MCS more elusive.11 _{Alternatively, since we know the actual interaction}

between the supply and demand with output and price, we could define the contem- poraneous relationship matrix B. It should be noted that when using the C-SVAR, as we do when choosing matrix B, we choose a C matrix with imposed restrictions that, based on theoretical conclusion, are assumed to be true. While the first option would allow the estimator to be tested along the lines of the empirical application in the fourth chapter, the second option does not increase the complexity of the

10_{Actually, all AR(2) coefficients, including a}2

12, will be zero, as neither output nor price were

modeled as AR(2) processes.

11_{If the interaction of price and output were explicitly modeled, this would impact on the per-}

formance of the remaining estimators which do not account for price in the risk-sharing estimate. On the other hand, if a specific output process is used solely for the SVAR-IV estimation, this would prohibit the cross-comparison of the estimators performance.

dynamics behind the DGPs which facilitates the interpretation of the MCS results. And so, the SVAR will be a B-model structure, where the true B matrix is known and supplied to the estimator, which in turn allows us, with the DGP’s at hand, to run SVARs that appropriately identify and separates output shocks into supply and demand shocks.12 _{The B-matrix is defined as}

B = " ψ 1 1 −1 #