# The Moorish Wanderer

## The Big Picture – Part 3

Posted in Dismal Economics, Moroccan Politics & Economics, Morocco, Read & Heard by Zouhair ABH on May 7, 2012

Looking back at the latest two posts, I must admit I had spent too much time trying to identify the deep parameters needed to build Morocco’s RBC model. But it was a blessing in disguise; indeed, when the standard methods of calibration are applied, the necessary structural parameters fit very well with the regressed/estimated results earlier; no harm done there.

This means a lot: calibrated parameters are usually constructed by means of steady state, and as far as academic modus operandi goes, there are no particular standard methods to follow (or in my case, by computing long-term trends for the variables of interest) – and the fact that econometric computations, however rudimentary and based upon a relatively small sample, tend to be vindicated by calibration method, does point out to the existence of some steady-state (that is yet to be determined) and the superiority of a long-term sample, compared to that of HCP’s forecast model.

The deep parameters’ vector encompasses the following values:

$\eta$ – time fraction allocated to labour activities : 1/3

$\alpha$ – Capital share output : .03414 (homogeneity of degree one is acquired, and Labour share is thus $1-\alpha$

$\theta$ – a measure for risk aversion : .0370

$\delta$ – depreciation rate of Capital (annual) : 2.905%

$\beta$ – household discount rate : .9198

$\rho$ – persistence of structural exogenous shocks: .923611

$\sigma$ – ‘white noise’ standard deviation : .00177

These figures are computed from stead-state identities:

1/$\alpha=\left[\frac{\delta+\frac{1}{\beta}-1}{\delta}\right]\frac{i_{lt}}{y_{lt}}$

2/$\theta=\frac{\delta+\hat{\beta}}{\delta}$ with $\beta=\frac{1}{1+\hat{\beta}}$

3/$\eta=\theta^{\alpha}$

4/$\phi=\frac{1}{1+\frac{2(1-\alpha)}{c_{lt}/y_{lt}}}$

Finally, the remaining figure to compute is $k_0$ which is derived from Capital motion law at steady state, with

$\frac{k_{lt}}{y_{lt}}=\frac{i_{lt}}{\delta y_{lt}}$

The subscript ‘lt’ refers to the steady-state proxy, i.e. long-term mean. As for the initial stock of capital (a figure nowhere to be found I am afraid) the idea is to use properties of the balanced growth path, with $y_0$ as the initial state for output, thus ensuring a ‘calibrated’ $k_0$

While these figures are not strictly equal to the estimates described before, they do fit in all of 95% Confidence Interval for each of their estimates, in fact they fit for all of them in the 99% CI. The simple difference being here that these parameters are fixed values, while estimates provided in the two last posts are in essence random variables, centred around the OLS estimates; it will be helpful in estimating the persistence factor of exogenous shocks. Indeed, the standard recipe to compute it is to consider it as the formula most able to capture the Solow Residual, defined such:

$\log(z_t)=\alpha\log k_t+(1-\alpha)\log n_t$

recall the first post for the log-linear argument; exogenous shocks are thus defined as AR(1) process, with a fixed term that denotes of long-term shocks;

$\log(z_t)=\rho\log(z_{t-1}) + (1-\rho)\log(z_{lt}) + \epsilon_{t-1}$

the next step is to compute $z_t$ just like all other aggregates with HP-filters, then regress the trend -i.e. $\log(z_{lt}$– on a index time to find $(1-\rho)$ which yields the following results:

      Source |       SS       df       MS              Number of obs =      62
-------------+------------------------------           F(  1,    60) = 2753.20
Model |  115.862298     1  115.862298           Prob > F      =  0.0000
Residual |  2.52497013    60  .042082836           R-squared     =  0.9787
Total |  118.387268    61  1.94077489           Root MSE      =  .20514
------------------------------------------------------------------------------
HP_z_t_sm_1 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
index |    .076389   .0014558    52.47   0.000     .0734769    .0793011
_cons |   22.80177   .0527426   432.32   0.000     22.69627    22.90727
------------------------------------------------------------------------------

The reader can observe that the coefficient .076389 means parameter rho is therefore has a value of .923611. Furthermore, the model’s mean (MS) square residual also allows for ‘white noise’ estimate, a zero-centred normal distribution with $\sigma=.00177$

We propose the following system of identities to solve Morocco’s RBC:

Households maximize and smooth their utilities over time according to the following:

$U\left(c_t;1-h_t\right)=E\sum\limits_{0}^{\infty}\beta^t\left[\frac{c_t^\gamma(1-h_t)^{(1-\gamma)}}{1-\phi}\right]^{1-\phi}$

the function is maximized subject to the following constraints:

$k_{t+1}=i_t+(1-\delta)k_t$

$y_t=z_t k_t^\alpha n_t^{1-\alpha}$

$i_t=s_t+\exp(z_t)tb_t$

$tb_t$ refers to the trade balance, the aggregate explanation for discrepancies between domestic savings and actual investment.

(next piece will deal with a first set of simulation and comparison with HP-filtered data in the first post)