© February 7, 2021, Christopher D. Carroll Equiprobable

An Equiprobable Approximation to the Bivariate Lognormal

Economic agents face risks of many kinds, which may mutually covary. A stock broker, for example, is likely to earn a salary bonus that is positively related to the performance of the stock market; if that broker also has personal stock investments, his financial wealth and labor income will be positively correlated.

The first part of this handout presents a convenient (and empirically realistic) formulation in which a consumer faces two shocks (which can be interpreted as a shock to noncapital income and a shock to the rate of return) that are distributed according to a multivariate lognormal that allows for correlation between them. The second part describes a computationally simple and convenient method for approximating that joint distribution.

1 Theory

Consider a consumer who faces both a risk to transitory noncapital income1

𝜃1,t+1 ≡  log Θ1,t+1 ∼  𝒩  (− 0.5 σ21,σ21)

and a risky log rate-of-return that is affected by following factors: the riskless rate r  ; a risk premium φ  ; an additional constant ζ  (whose purpose will become clear below); a component that is linearly related to 𝜃
 1,t+1   ; and an independent shock                  2   2
𝜃2 ∼  𝒩  (− 0.5σ 2,σ 2)  :

r     ≡ log R      =  r + φ +  ζ +  ω𝜃     (σ  ∕σ  ) + 𝜃
  t+1           t+1                      1,t+1   2   1     2,t+1

for some constant ω  . Since (σ2 ∕σ1 )ω𝜃1,t+1   is the only component of rt+1   that covaries with 𝜃
 1,t+1   ,

cov (𝜃     ,r    ) = cov (𝜃     ,(σ  ∕σ  )ω 𝜃     )
      1,t+1   t+1            1,t+1    2   1    1,t+1
                  =  ω (σ2 ∕σ1 )cov (𝜃1,t+1,𝜃1,t+1)
                                ◟-------◝ ◜-------◞

                  =  ω σ2 σ1.

Equation (2) yields a description of the return process in which the parameter ω  controls the correlation between the risky log return shock and the risky log labor income shock. If ω  =  0  the processes are independent.

Now we want to find the value of ζ  such that the mean risky return is unaffected by   2
σ 1   (so that we will be able to understand clearly the distinct effects of labor income risk, the independent component of rate-of-return risk   2
σ 2   , and the correlation between labor income risk and rate-of-return risk, ω  ). Thus, we want to find the ζ  such that

𝔼t[Rt+1  ] = e

regardless of the values of σ2
 1   and σ2
 2   . We therefore need:

    𝔼 [e                     ] = 1.
log 𝔼 [eζ+(σ2∕σ1)ω𝜃1,t+1+𝜃2,t+1] = 0.

Using standard facts about lognormals (cf. MathFacts), and for convenience defining ˆω  = ( σ ∕σ  )ω
        2   1  , we have

0.=   ζ − 0.5 ˆω σ2 −  0.5σ2 +  0.5ωˆ2 σ2  + 0.5 σ2
                 1         2           1        2
   =  ζ − 0.5 σ2ωˆ(1 −  ˆω )
               1 2   2                    2  2
 ζ =  0.5(ˆω  − ˆω  )σ 1 = 0.5 (ω σ2σ1 −  ω  σ2 ).

2 Computation

A key step in the computational solution of any model with uncertainty is the calculation of expectations. Writing ˜      ˜
Θ1  ≡  Θ1,t+1   and  ˜
R  ≡  Rt+1   and 𝔼 [∙ ] = 𝔼t [∙t+1 ]  , the expectation of some function h  that depends on the realization of the risky return  ˜
R  and the labor income shock is:

                 ∫  ¯Θ1 ∫ R¯
𝔼 [h (˜Θ  ,R˜ )] =            h(˜Θ  ,R˜ )dF (˜Θ  ,R˜ )
       1                        1           1
                  Θ1    R-

where F (˜Θ1, ˜R )  is the joint cumulative distribution function. Standard numerical computation software can compute this double integral, but at such a slow speed as to be almost unusable. Computation of the expectation can be massively speeded up by advance construction of a numerical approximation to F (˜Θ1, R˜ )  .

Such approximations generally take the approach of replacing the distribution function with a discretized approximation to it; appropriate weights wi,j  are attached to each of a finite set of points indexed by i  and j  , and the approximation to the integral is given by:

                   n   m
                 ∑    ∑
𝔼 [h(Θ˜1,  ˜R )] ≈          h (ˆΘ1 [i,j],Rˆ[i,j])w [i,j]
                  i=1 j=1

where the  ˆ
Θ1   and  ˆ
R  matrices contain the conditional means of the two variables in each of the {i,j } regions. Various methods are used for constructing the weights w [i,j]  and the nodes (the i  and j  points for Θ1   and R  ).

Perhaps the most popular such method is Gauss-Hermite interpolation (see Judd (1998) for an exposition, or Kopecky and Suen (2010) for some alternatives). Here, we will pursue a particularly intuitive alternative: Equiprobable discretization. In this method, m  =  n  and boundaries on the joint CDF are determined in such a way as to divide up the total probability mass into submasses of equal size (each of which therefore has a mass of n − 2   ). This is conceptually easier if we represent the underlying shocks as statistically independent, as with 𝜃
 1,t+1   and 𝜃
  2,t+1   above; in that case, each submass is a square region in the Θ1   and Θ2   grid. We then compute the average value of Θ
  1   and R  conditional on their being located in each of the subdivisions of the range of the CDF. Since, in this specification, R  is a function of Θ1   , the R  values are indexed by both i  and j  , but since we have written Θ
  1   as IID, the representation of the approximating summation is even simpler than (7):

                     ∑ n  ∑n
𝔼 [h (˜Θ  ,R˜) ] ≈ n − 2        h (ˆΘ  [i], R (ˆΘ  [i], ˆΘ [j]))
       1                           1         1      2
                      i=1 j=1

where the function R (Θ1, Θ2 )  is implicitly defined by (2).

Details can be found in the Mathematica notebook associated with this handout. A particular example, in which σ22 =  σ21   and ω  = 0.5  , is illustrated in figure 1; the red dots reflect the height of the approximation to the CDF above the conditional mean values for Θ1   and R  within each of the equiprobable regions.

Figure 1:‘True’ CDF With Approximation Points in Red for ω =  0.5



   Judd, Kenneth L. (1998): Numerical Methods in Economics. The MIT Press, Cambridge, Massachusetts.

   Kopecky, Karen A., and Richard M.H. Suen (2010): “Finite State Markov-Chain Approximations To Highly Persistent Processes,” Review of Economic Dynamics, 13(3), 701–714, http://www.karenkopecky.net/RouwenhorstPaper.pdf.