Efecto de la ventilación-climatización en las condiciones higrotérmicas de las bodegas

In the estimation described below, we allow measurement errors on all the CDS matu- rities. This estimator diers from the Maximum Likelihood a la Chen and Scott (1993) because we do not use the non-central chi-squared distribution and we do not restrict the measurement errors to follow a normal distribution. In fact, the likelihood function remains relatively simple and tractable.

timation. Therefore, we believe that the Kalman lter estimation will give us a good robustness check on the results discussed above.

6.8.1 State Space Representation:

The model can be expressed in the state space form by adding measurement errors to the equation of the observable CDS prices.

ht=a+ Φ∗ht−1+vt (6.35)

−lnQt=−lnA(t, si) +B(t, si)∗ht+εt (6.36)

where ε0_t = (ε1, ε3, ε5, ε7, ε10). εi represents the measurement error term introduced

to allow for imperfections and small errors in each maturity i. Since ht follows a CIR

process, we can rewrite it as follows:

ht =µ1(1−exp(−α1∗ 4t)) +exp(−α1∗ 4t)∗hj,t−1+v1t (6.37)

−lnQt=−lnA(t, si) +B(t, si)∗ht+εt (6.38)

where i= 1, ....,5 represents the number of maturities considered.

The error term vt represents the unanticipated change (or innovation) in the state

variableht and has a conditional expected mean of zero and conditional variance equal

to: var(vt|vt−1) =σ1∗ 1−exp(−α1∗ 4t) α1 (0.5∗µ1∗(1−exp(−α1∗4t))+exp(−α1∗4t)∗ht−1 (6.39)

We adopt the standard assumption in Kalman lter and we assume that there is no serial or cross sectional correlation between the measurement errors. Therefore, we can write the covariance matrix as follows:

Et−1   vt εt     vt εt   0 =   Ht 0 0 U   (6.40)

whereHtand U are the conditional variances of the state variable and the variances

of the measurement errors on the diagonal respectively.

6.8.2 Quasi Maximum Likelihood (QML) Estimation:

The xed parameters of the state space representation described above are typically estimated using the maximum likelihood method. In our case, we are assuming that

ht and yt are following a CIR process, therefore the innovations of the state variables

are not normally distributed which make the standard linear Kalman lter a biased estimator of the unobservable state variables. However, De Jong (2000) and Chen an Scott (2002) perform Monte-carlo exercise and show that there is no evidence of signicant bias in the parameter estimates. Therefore, by using the QML estimator the authors prove that it is possible to obtain consistent estimators. We follow their approach and estimate the parameters using the QML as follows:

maxθlnL= T X t=1 Lt = T X t=1 −0.5∗(ln|Ht|+u 0 t∗F −1 t ∗ut) (6.41)

whereθrepresents the vector of parameters to optimizeθ= (α1, µ1, σ1, λ1, ε1(1), ε1(3), ε1(5), ε1(7), ε1(10))

and ut and Ft are dened as

Ft=E(ut∗u 0

t) (6.43)

The estimation is achieved by calculating recursively the distribution of the state variablehtconditional on the observations at time t. We start the recursion by assuming

that the initial values at time zero ofh0andH0 are equal to the conditional meanµ1and variance µ1∗(σ1)

2

2∗α1 respectively. The Kalman lter algorithm consists of three important

steps: prediction, updating and estimation. In order to estimate the parameter vector

θ, a good initial guess is of a key importance. Therefore, we try dierent initial values

until we reach the global maximum and we estimate the standard errors using the White method (1982).

For a more detailed description of the Kalman lter estimation applied to the CIR process, we refer to De Jong (2000) and Chen and Scott (2002).

6.8.3 Kalman Filter Results:

Table 6.5, shows that the new introduced errorε1(5)is signicant across all the countries and during both the pre-crisis and crisis period, which supports our idea of introducing a measurement error for the 5 year maturity. We can also see that the credit premium

λ1 is signicant for the period preceding the crisis for Brazil and Turkey. However λ1 has a weak eect during the crisis period. Similarly to the previous section, Table 6.5 implies strong mean reversion (i.e. low half life values), in fact it seems that the speed of the mean reversion is higher than what was reported in Table 6.3.

Concerning the liquidity risk, we can see in Table 6.6 that the evidence of existence of a liquidity premium is stronger than that of ML a la Chen and Scott (1993). The table also shows that the liquidity premiumλ2 is stronger during the pre-crisis than the crisis period. In our model, we assumed that liquidity risk enters only the defaultable leg of the CDS contract which means that the protection buyer will pay a lower price

if the CDS market is illiquid15_{, this is equivalent to say that the protection buyer will}

earn the liquidity premium16_{. Table 6.6 reveals that during the pre-crisis period the}

liquidity premium is positive and signicant for all the countries, this indicates that the price of liquidity risk has signicant impact on the defautlable leg of the CDS contract and does reduce CDS spreads in line with our model assumption. However, we observe that liquidity premium λ2 decreases in magnitude and even disappears during crisis time. In fact, this situation is possible if, in crisis time, there is a high demand of CDS protection to hedge against default risk. If the demand of protection is strong then protection buyers are not able to demand any compensation for liquidity risk because the CDS market is more liquid. Therefore, protection sellers are able to benet from the shift in the demand and supply to either lower the liquidity premium earned by protection buyers (i.e. case of Brazil, we observe that λ2 has low signicance level) or to remove it (i.e case of Philippines and Turkey, we observe thatλ2 is insignicant).