This paper proposes the use of an estimator calculated using the generalized method of moments continuously updated to characterize a linear stochastic discount factor for a given economy. The estimator is applied to the Mexican and Chilean stock markets for 2007-2012, this period includes the international financial crisis. The stochastic discount factor, for both economies, took values of less than one and presented high market volatility values during several years. A comparison with the results from the two stages generalized methods of moments and the iterative one is also discussed, showing the superiority of the continuous updating estimator over these two frequently used estimation techniques

Se propone utilizar el estimador calculado por el método de momentos generalizado continuamente actualizado para caracterizar el factor de descuento estocástico de una economía. El estimador se aplica a los mercados accionarios de México y Chile en el período 2007-2012, que incluye el período de la crisis financiera internacional, en el cual en ambas economías el factor de descuento estocástico muestra años en los cuales fue menor que uno y la volatilidad del mercado fue alta. Se compara y discuten los resultados del método generalizado de momentos de dos etapas y los del iterativo, y se muestra la superioridad del estimador continuamente actualizado sobre estas dos técnicas de estimación tan usadas.

The stochastic discount factor is extensively quoted in financial literature when referring to risk adjustments. This article proposes the use of the continuously updated estimator to identify the linear stochastic discount factor. The estimator is applied to assess the changes in the stochastic discount factor in the Mexican and Chilean economies during the period 2007-2012, which includes the period of the international economic credit crisis 2008-2009.

The main applications of the stochastic factor are in asset pricing theory, in valuations and in the assessment of market efficiency.

The paper is divided in four sections: Section one, which introduces the moment conditions starting from a simple representative consumer- investor problem, section two, which gives an overview of the Mexican and Chilean economies during the period studied, section three, which includes the analysis and discussion of the empirical results. Finally, the conclusions section follows.

Considering that a consumer can freely trade assets i, and that the expected value of a discounted time-separable utility is maximized,

where the subjetive discount factor δ measures the personal time preference, 0 < δ < 1, C_{
t+j
} is the investor's consumption in period t + j, and U(C_{t+j}) is the
period utility of consumption at

where w_{i,t}, is the proportion invested in risky asset_{
ft
} is the return of risky asset i in period t and _{
i t
} is the return of the risk free asset in period

The optimal consumption and portfolio plan must be such that satisfies that the marginal
utility of consumption today is equal to the expected marginal utility benefit from
investing one monetary unit in asset _{
i ,t+1
} and consuming the proceeds, conditional on ,Ѱ which is a subset of the
available information at t, A_{t}

If both sides are divided by U'(Ct), then

where the stochastic discount factor m_{
t+1
} is equal to the stochastic intertemporal rate of substitution
δU'(C_{t+l}) / U'(C_{
t
} ).

Notice that if the returns of the n risky assets in the economy are the vector R_{t}, and 1 is a vector of ones, relationship (4) can be written as

where R_{
t
} has an unconditional non-singular variance-covariance matrix Σ.

An implication of this model and other inter-temporal asset pricing ones is that

where the return on one period riskless bond is

For example, in the case of power utility,

If the information set is normal, any payoff satisfies

which can be written as:

If the One Factor Capital Asset Pricing Model is satisfied,

where γ is a benchmark's risk premium, in equilibrium, the market return minus the risk free return.

Assume that the stochastic discount factor m_{
t
} has the form

In equilibrium, the conditional moment condition the stochastic discount factor m_{
t
} must satisfy, conditioned to previous information Ψ_{t-1} is that the expected product of any return R_{
t
} considering the discount factor must be equal to one,

In particular, deviations in the moment condition can be interpreted as return's alpha for the investor, as in

The Euler equation of consumption (14) shows the expected rate of return on the assets as well as relative expected consumption stream which is negatively related to the risk aversion parameter.

This shows whether consumers prefer to trade-off their current consumption for higher consumption levels in the future. In order to estimate preference parameters of the Euler equation, the constant relative risk aversion coefficient (CRRA) y and discount factor δ, the GMM technique is used. The necessary condition for the GMM method to estimate the structural parameters is that the moment must hold.

To get the moment condition from

According to

where x_{
t
} is a vector of variables observed by agents at time t and β_{0} is a p dimensional parameter vector to be estimated. Therefore:

In general, let us construct an objective function that depends only on the available information of the agents and unknown parameters β. Let

The value of g_{
T
} (_{
0
} should be close to zero for large values of T. This paper, follows _{T},

where W_{
T
} is a symmetric, positive definite weighting matrix. W_{
T
} can be estimated minimizing

The weighting matrix W_{
T
} is chosen so that g_{
T
} is close to zero, taking into account possible heteroscedasticity and autocorrelation (HAC) behavior.

The GMM estimator does not require the specification of the joint distribution of the observed variables, unlike the maximum likelihood (ML) estimator.

The instrument vector needs to be predetermined in the period when the agent forms his expectations. Both past and present values of the variables in the model can be used as instruments. The model estimator is consistent even when the instruments are not exogenous or when the disturbances are serially correlated.

The iterated generalized method of moments estimator is calculated as follows: To compute W_{
T
} a consistent estimator of β_{
0
} is needed. This can be obtained by initially using W_{
T
} = l_{
rxr
} (identity matrix) and suboptimal choice of βin minimizing J_{
T
} (β) (18) and obtaining, therefore, the values of β_{
T
} . By using this value of β in (19), W_{
T
} is obtained. Again, by using the new values of W_{
T
} , β_{
T
} can be obtained by minimizing equation (18). This process is repeated until the estimates converge. According to

Furthermore, the continuous updating estimator (CUE), proposed by

In the period of study five sup-periods can be identified: a slowdown of the economy, during 2007 and 2008, the crisis in Mexico, at the end of 2008 and beginning of 2009, the recovery period, 2009, 2010 and 2011 and a slowdown of the economy, at the end of 2012. During 2007, the economy slowed down because of the credit crises in the United States weakened its economy, Mexican exports moderated their growth and commodity prices increased: oil, food and metallic supplies. In august of 2008, the international banking market crises aggravated. With the bankruptcy of Lehman Brothers in September, uncertainty in the international market grew. The international markets lacked liquidity. The crises expanded to other financial markets, including the Mexican one. By the second quarter of 2008, the crisis effects began to subside. The actions that Mexican and international authorities had implemented started to give results. Progressively, market liquidity increased, the uncertainty diminished and the growth returned to the Mexican economy. During 2012, the uncertainty derived from the European Crisis affected the American Economy. Mexican exports slowed down and the manufacturing activity in some regions of the country, contracted. These were signs of a possible deterioration of the economic activity prospects in México.

The Chilean government conducts a rule-based countercyclical fiscal policy, accumulating surpluses in sovereign wealth funds during periods of high copper prices and economic growth, and allowing deficit spending only during periods of low copper prices and growth.

As a result, Chile had a mild economic crisis as a consequence of the world wide credit crises. Chile benefited from a governmental rule-based countercyclical fiscal policy. The economics went from a recovery period in 2006 and 2007 a to slowdown period in 2008. Chile only suffered the world wide crises consequences in 2009. By 2010, the Chilean economy was fully recovered. During the period of 2010 to 2012, it grew 6%, each year. In 2012, in spite of the European crises, the Chilean economy kept growing.

Inflation decreased gradually during 2006 to 2008, from being 13% in 2006 to 5% and 1% 2007 and 2008, respectively. In 2009 and 2010, as a consequence of the international economic crisis and the contra cyclical expansionary measures, inflation rebounded to 4% and 7%, respectively. For 2011 and 2012, prices stabilized, inflation grew only 3% and 2%, respectively.

This study analyses the performance of the Mexican Stock Market and the Chilean Stock Market. In each one a market index is selected as benchmark. The index used in the Mexican Market was the Total Return Index "índice de Rendimiento Total (IRT)” and for the Chilean Market, the Santiago Stock Exchange Index "índice de la Bolsa de Santiago "IPSA”, both indexes are cash dividends adjusted. The mean and standard deviation of these indexes during the period under study are shown in

Country Index
Chile: IPSA
México: IRT
Year
Mean
Std. Dev.
Mean
Std. Dev.
2007
1.000581
0.012185
1.00062
0.01352
2008
0.999169
0.01848
0.999252
0.022944
2009
1.001694
0.010248
1.00166
0.017061
2010
1.001304
0.007358
1.000819
0.009072
2011
0.999442
0.013889
0.999994
0.01233
2012
1.000136
0.005965
1.000741
0.007104

Daily returns.

Source: Own elaboration

Mexico
Chile
Year
Mean
Std. Dev.
Mean
Std. Dev.
2007
1.001546
0.0220236
1.001008
0.0209549
2008
0.9989916
0.036032
0.9987378
0.027021
2009
1.002161
0.0324366
1.002126
0.0274227
2010
1.001043
0.0196362
1.002244
0.0282331
2011
1.000147
0.0228024
0.9997626
0.036385
2012
1.001278
0.0190305
1.000299
0.026818

Based on assets with at least 60 quotes in the year.

Source: Own elaboration

If

where _{
t
} can be written as

The two parameter model follows from

México
Chile
2007
retine
-21.82258
-3.62
***
-33.32382
-10.12
***
cons
0.0031768
5.58
***
0.0031076
12.55
***
2008
retine
-26.54003
-6.38
***
-14.78938
-8.56
***
cons
0.0067242
4.6
***
0.0014661
4.73
***
2009
retine
490.553
2.54
**
-470.0749
-3.45
***
cons
-0.0825211
-2.46
**
0.0236357
3.64
***
2010
retine
-73.860
-2.81
***
-60.10842
-3.35
***
cons
0.0042448
3.43
***
0.0032123
6.57
***
2011
retine
-9.780145
-1.97
**
-28.54276
-8.69
***
cons
0.0007979
1.73
*
0.0024213
6.32
***
2012
retine
-71.010
-2.5
**
-229.3429
-4.87
***
cons
0.00293
3.96
***
0.003834
4.46
***

*** ** * statistically significant at the 99%, 95% and 90%.

Source: Own elaboration

The anomalous results in 2009 for Mexico are associated with overidentification in the model. The probability that the Chi square value of the Hansen overidentification test be equal to zero is 0.02, see

Mexico
Chile
Hansen J statistic
Chi-sq(l) P-val
Hansen J statistic
Chi-sq(l) P-val
2007
3.502
0.0613
3.077
0.0794
2008
1.354
0.2446
3.957
0.0467
2009
5.412
0.02
11.358
0.0008
2010
0.039
0.8439
9.199
0.0024
2011
0.002
0.9683
14.465
0.0001
2012
1.63
0.2018
0.227
0.6339

Source: Own elaboration

For the Chilean economy, in the two parameter model, the discount factor is proportional to the market index and negative, and it is possible an alpha excess return over the one market factor model. Alphas are positive in all periods and statistically significant. The slope coefficients of the return are negative and statistically significant. There is no observable sign change in the discount factor model for the period under study.

However, the increase in the sensibility of the discount factor to the market index during 2009 (the recovery period) and 2012 (the European crisis period), this contrasts with the small beta coefficient in 2008 (the international credit crisis) is noticiable, this contrasts with, see

Mexico
Chile
ret_me
Coef.
Z
Coef.
z
2007
-27.027
-3.8
* **
-36.952
-10.24
** *
2008
-23.708
-6.66
* **
-14.622
-8.66
** *
2009
99.838
5.96
* **
200.184
7.67
** *
2010
-263.009
-1.58
-652.440
-3.16
** *
2011
-9.347
-1.99
**
-26.822
-9
** *
2012
-82.225
-2.47
**
-229.048
-4.77
** *

Source: Own elaboration

In the restricted one coefficient model, similar results are observed. In the Mexican model, all beta coefficients are negative, except for 2009, for which the Hansen non-overidentification hypothesis cannot be accepted. In 2010 and 2012, there is an augmented sensitivity of the stochastic discount factor to the market index. However, during 2012 the Hansen non-overidentification hypothesis is rejected. In the Chilean model, the results are similar to those observed in the Mexican model. However, the beta coefficient has a positive sign for the year 2009, which can be due to overidentification, see

Mexico Hansen J statistic
Chi-sq(l) P-va
Chile Hansen J statistic
Chi-sq(l) P-va
2007
2.587
0.1077
2.084
0.1488
2008
2.78
0.0954
3.55
0.0596
2009
16.538
0
33.819
0
2010
0.597
0.4398
8.045
0.0046
2011
0.006
0.9387
14.865
0.0001
2012
12.744
0.0004
0.6842
0.6842

Source: Own elaboration

The analysis using the two step GMM and IGMM methods shows less reliable estimates than the CUE estimator, see

México
Chile
Two steps
Coef.
Z
Coef.
z
2007
-15.446
-1.22
117.514
6.89 ***
2008
-66.896
-3 37 ***
-78.718
-2.24 **
2009
35.871
2 74 ***
55.959
23 **
2010
-5.584
-0.09
187.067
2.47 **
2011
-4.412
-0.24
9.572
0.42
2012
79.108
2.56 **
106.985
0.83
Igmm
Coef.
Z
Coef.
z
2007
-15.646
-1.24
116.922
59\ ***
2008
-65.927
-3 42 ***
-82.058
-2.28 **
2009
36.088
2 75 ***
60.324
2.42 **
2010
-42.257
-0.67
194.799
2.52 **
2011
-5.030
-0.27
13.211
0.59
2012
78.831
2.55 **
92.930
0.75

The first three lags of the excess market return were used as instruments. * ** *** statistically significant at 90, 95 and 99 percent.

Source: Own elaboration

The applications show that the stochastic discount factor changed during the previous crisis credit period, in years 2009 and 2010, in the Mexican and Chilean economies. In 2009 the sensitivity to the index became abnormally positive. In 2010, it became abnormally large, although negative.

Using the continuous updating estimator (CUE), the alpha for Mexico and Chile is positive in all years except for 2009, when there are over identification issues. For Mexico and Chile, in all years except for Mexico in 2007 and 2012, the betas are negative, that is, the discount factor is inversely related to IRT factor.

The sizes of the betas are related to the economics and economic policies implemented in the countries. In Mexico and Chile, betas in absolute terms were higher during the recovery period from the credit crisis and during the European crisis and lower during the credit crisis.

The results suggest that using a continuous updating estimator gives more reliable estimates of the linear stochastic discount factor than the two stages or the iterated general method of moments estimators, particularly if instruments are weak.